Analysis & Linear Algebra

Continuity and Differentiation

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1. Continuity, uniform continuity, Lipschitz

Read this as a chapter →

Lec-25 Uniform Continuity

NPTEL · Real Analysis (S. H. Kulkarni, IIT Madras)

Uniform continuity and its relation to compactness (Heine–Cantor).

Lec-31 Uniform Continuity and Absolute Continuity

NPTEL · A Basic Course in Real Analysis (IIT Kharagpur)

Worked examples of proving/disproving uniform continuity.

Continuity vs uniform continuity vs Lipschitz free

In plain words

Continuous means small input changes give small output changes, with how small depending on where you stand. Uniform means one answer works everywhere. Lipschitz puts an actual number on the steepness.

The chain

Lipschitz ──▶ uniformly continuous ──▶ continuous
    ✗ ◀──── (√x on [0,1])      ✗ ◀──── (1/x on (0,1), x² on ℝ)

Heine–Cantor: continuous on a compact set ⇒ uniformly continuous. This is the single most-used theorem here.

Portable tests

Bounded derivative ⇒ Lipschitz (mean value theorem). So sin, cos, arctan are Lipschitz on $\mathbb{R}$ .
Uniformly continuous on a bounded set ⇒ bounded. So 1/x is not uniformly continuous on (0,1).
Uniformly continuous ⇒ maps Cauchy sequences to Cauchy sequences. Spoiler for $1/x: x_{n} = 1/n$ is Cauchy, $f(x_{n}) = n$ is not.
Uniformly continuous on (a, b) ⇔ extends continuously to [a, b]. sin(1/x) on (0,1): no limit at 0 ⇒ not uniformly continuous.
Unbounded domain: f uniformly continuous on $[0, \infty) \Rightarrow$ |f(x)| ≤ A + B|x| (at most linear growth). So $x^{2}, e$ ˣ fail; $\sqrt{x}, \sin(x^{2})$ ? — $\sin(x^{2})$ is bounded yet not uniformly continuous (oscillates faster and faster: points $\sqrt{2\pi{}n}$ and $\sqrt{2\pi{}n + \pi/2}$ get arbitrarily close but values differ by 1).
Sum/product: sums of uniformly continuous functions are uniformly continuous; products need not be $(x \cdot x = x^{2}$ on $\mathbb{R})$ .

The standard six

f	Domain	Continuous	Unif. cont.	Lipschitz
$\sqrt{x}$	[0, 1]	✓	✓ (compact)	✗
1/x	(0, 1)	✓	✗	✗
$x^{2}$	$\mathbb{R}$	✓	✗	✗
$x^{2}$	[0, 1]	✓	✓	✓
sin(1/x)	(0, 1)	✓	✗	✗
$\sin(x^{2})$	$\mathbb{R}$	✓	✗	✗

Discontinuity types worth knowing

Thomae's function: continuous at irrationals, discontinuous at rationals (Riemann integrable).
Dirichlet $1_\mathbb{Q}$ : discontinuous everywhere.
Monotone functions have only jump discontinuities, countably many.
A function continuous exactly on $\mathbb{Q}$ cannot exist (continuity set is $a G\delta; \mathbb{Q}$ is not).

Key takeaways

Compact domain turns continuity into uniform continuity for free.
To kill uniform continuity: find two sequences with | $x_{n} - y_{n}$ | → 0 but | $f(x_{n}) - f(y_{n})$ | ↛ 0.
Bounded does not imply uniformly continuous $(\sin(x^{2}))$ ; uniformly continuous does not imply bounded (x on $\mathbb{R})$ .

The ε–δ limit machineinteractivefree

Exam focus: The chain Lipschitz ⇒ uniformly continuous ⇒ continuous, where each converse fails, and what compactness or boundedness of the domain changes. Know √x, 1/x, x², sin(1/x), sin(x²) by heart.

Practice

Part C2025 Decembershow ▾

Consider the following real-valued functions

F_{1}

and

F_{2}

defined on

\mathbb{R}

, given by

F_{1}(x)=1

if x≥1, 0 otherwise, and

F_{2}(x)=\int_{0}

ˣexp(−t)dt if x≥0, 0 otherwise. Define another function

F:\mathbb{R}\to[0,1]

F(x)=(2/3)F_{1}(x)+(1/3)F_{2}(x)

for all

x\in\mathbb{R}

. Which of the following statements are true?

A.F is non-decreasing on $\mathbb{R}$ ✓ Correct
B. $\lim(x\to\infty) F(x) = 1$ ✓ Correct
C.F is left-continuous on $\mathbb{R}$
D.F is right-continuous on $\mathbb{R}$ ✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2025 Decembershow ▾

Let

f : (0,\infty) \to (0,\infty)

be a uniformly continuous function. Define

g : (0,\infty) \to (0,\infty)

g(x) = x^{2}

if 0 < x ≤ 1, and g(x) = x if x > 1. Which of the following statements is true?

A.f∘g is uniformly continuous, but g∘f is not uniformly continuous.
B.g∘f is uniformly continuous, but f∘g is not uniformly continuous.
C.Neither f∘g nor g∘f is uniformly continuous.
D.Both f∘g and g∘f are uniformly continuous.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾

Let

f : \mathbb{R} \to \mathbb{R}

be a non-constant continuous function. Which of the following statements is necessarily true?

A.For every bounded subset $A \subseteq \mathbb{R}, f$ ⁻ $^{1}(A)$ is a bounded subset of $\mathbb{R}$ .
B.For every Cauchy sequence $(x_{n})_{n}\ge_{1}$ in $\mathbb{R}, (f(x_{n}))_{n}\ge_{1}$ is a Cauchy sequence in $\mathbb{R}$ .✓ Correct
C.There exists $x \in \mathbb{R}$ such that f(x) = x.
D.There exists $x \in \mathbb{R}$ such that f(x) = 0.

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Juneshow ▾

Let S be a dense subset of

\mathbb{R}

and

f : \mathbb{R} \to \mathbb{R} a

given function. Define

g : S \to \mathbb{R}

by g(x) = f(x). Which of the following statements is necessarily true?

A.If f is continuous on the set S, then f is continuous on the set $\mathbb{R}$ \ S
B.If g is continuous, then f is continuous on the set S
C.If g is identically 0 and f is continuous on the set $\mathbb{R}$ \ S, then f is identically 0✓ Correct
D.If g is identically 0 and f is continuous on the set S, then f is identically 0

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

Let

f : [0, \infty) \to [0, \infty)

be a continuous function such that

\lim(x\to\infty) f(x)/x = \alpha < 1

. Let S = {

c \in [0, \infty)

| f(c) = c}. Which of the following statements are true?

A.S is empty.
B.S is non-empty.✓ Correct
C.f must be uniformly continuous.
D.f need not be uniformly continuous.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

Let

f : \mathbb{R} \to \mathbb{R}

be a continuous function such that f(x) = 0 for all x ≤ 0 and for all x ≥ 1. Define

F(x) = \sum(n=-\infty..\infty) f(x + n), x \in \mathbb{R}

. Which of the following statements are true?

A.F is bounded.✓ Correct
B.F is continuous on $\mathbb{R}$ .✓ Correct
C.F is uniformly continuous on $\mathbb{R}$ .✓ Correct
D.F is not uniformly continuous on $\mathbb{R}$ .

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Juneshow ▾

What is the cardinality of the set of real solutions of eˣ + x = 1?

A.0
B.1✓ Correct
C.Countably infinite
D.Uncountable

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾

Let

f : \mathbb{R} \to [0, \infty)

be a bijective function. Which of the following statements is true?

A.f is monotone.
B.f is continuous but not strictly monotone.
C.f is not continuous.✓ Correct
D.f is continuous but not uniformly continuous.

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Juneshow ▾

Let

f : \mathbb{R} \to \mathbb{R}

be a continuous and one-to-one function. Which of the following statements are necessarily true?

A.f is strictly increasing
B.f is strictly decreasing
C.f is either strictly increasing or strictly decreasing✓ Correct
D.f is onto

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Juneshow ▾

Let

K \subseteq \mathbb{R}

be non-empty and f : K → K be continuous such that |x − y| ≤ |f(x) − f(y)| for all x, y ∈ K. Which of the following statements are true?

A.f need not be surjective✓ Correct
B.f must be surjective if K = [0, 1]✓ Correct
C.f is injective and f⁻ $^{1} : f(K) \to K$ is continuous✓ Correct
D.f is injective, but f⁻ $^{1} : f(K) \to K$ need not be continuous

The worked solution and the per-option analysis are in Trap Analysis.

Part B2023 Juneshow ▾

Which one of the following functions is uniformly continuous on the interval (0, 1)?

A.f(x) = sin(1/x)
B.f(x) = $e^{-1/x^{2}}$ ✓ Correct
C.f(x) = eˣ cos(1/x)
D. $f(x) = \cos x \cdot \cos(\pi/x)$

Solution

A continuous function on (0,1) is uniformly continuous iff it extends continuously to [0,1]. $e^{-1/x^{2}}$ → 0 as x → 0⁺, so it extends; the other three oscillate without a limit at 0.

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Part C2023 Decembershow ▾

Let

f : [0, \infty) \to \mathbb{R}

be the periodic function of period 1 given by f(x) = 1 − |2x − 1| for x ∈ [0, 1], and define

g : [0, \infty) \to \mathbb{R}

g(x) = f(x^{2})

. Which of the following statements are true?

A.f is continuous on $[0, \infty)$ .✓ Correct
B.f is uniformly continuous on $[0, \infty)$ .✓ Correct
C.g is continuous on $[0, \infty)$ .✓ Correct
D.g is uniformly continuous on $[0, \infty)$ .

Solution

f is a continuous tent function with f(0) = f(1) = 0, so its periodic extension is continuous; continuous periodic ⇒ uniformly continuous. g = f∘ $x^{2}$ is continuous but its oscillations speed up as $x \to \infty ($ the tents get compressed), so it is not uniformly continuous.

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Part C2023 Decembershow ▾

Let

f : \mathbb{R} \to \mathbb{R}

be a continuous function such that |f(x) − f(y)| ≥ log(1 + |x − y|) for all

x, y \in \mathbb{R}

. Which of the following statements are true?

A.f is necessarily one-one.✓ Correct
B.f need not be one-one.
C.f is necessarily onto.✓ Correct
D.f need not be onto.

The worked solution and the per-option analysis are in Trap Analysis.

Part C2023 Decembershow ▾

Which of the following statements are true?

A.f(x) = [x] sin(1/x) for x ≠ 0, f(0) = 0 (where [x] is the integer part) has a discontinuity at 0 which is removable.
B. $f : [0,\infty) \to \mathbb{R}, f(x) = \sin(\log x)$ for x ≠ 0, f(0) = 0, has a discontinuity at 0 which is NOT removable.✓ Correct
C.f(x) = $e^{1/x}$ for x < 0, f(x) = $e^{1/(x+1)}$ for x ≥ 0, has a jump discontinuity at 0.✓ Correct
D.If $f, g : [0,1] \to \mathbb{R}$ are of bounded variation, then fg has at most countably many discontinuities.✓ Correct

Solution

(1) As x → 0⁻, [x] = −1 and −sin(1/x) oscillates: no left limit, not removable.

(2) sin(log x) oscillates as x → 0⁺.

(3) Left limit 0, right limit e.

(4) A product of BV functions is BV, and BV functions have only countably many (jump) discontinuities.

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Part Bshow ▾

Let

f : \mathbb{R} \to \mathbb{R}

be uniformly continuous. Which of the following is necessarily true?

A.f is bounded
B.f is Lipschitz
C.|f(x)| ≤ A + B|x| for some constants A, B✓ Correct
D.f is differentiable almost everywhere

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾

Let

f : (0, 1) \to \mathbb{R}

be continuous. Which of the following are equivalent to f being uniformly continuous on (0, 1)?

A.f extends to a continuous function on [0, 1]✓ Correct
B.f is bounded
C.f maps Cauchy sequences to Cauchy sequences✓ Correct
D.f is Lipschitz

The worked solution and the per-option analysis are in Trap Analysis.

Part Bshow ▾

Let f : [0, 1] → [0, 1] be continuous. Which of the following must be true?

A.f has a fixed point.✓ Correct
B.f is differentiable on (0, 1).
C.f is monotone.
D.f is Lipschitz.

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾

Let

f : (0, 1) \to \mathbb{R}

be continuous. Which of the following are true?

A.f is bounded
B.If f is uniformly continuous then f is bounded✓ Correct
C.If f is Lipschitz then f is uniformly continuous✓ Correct
D.f extends continuously to [0, 1]

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾

Which of the following functions are uniformly continuous on the given domain?

A. $f(x) = \sqrt{x}$ on $[0, \infty)$ ✓ Correct
B. $f(x) = \sin(x^{2})$ on $\mathbb{R}$
C.f(x) = x sin(1/x) on (0, 1)✓ Correct
D.f(x) = eˣ on $[0, \infty)$

Solution

(a) Hölder-½ with | $\sqrt{x} - \sqrt{y}$ | $\le \surd$ |x − y|.

(b) oscillation speeds up: not uniformly continuous.

(d) super-linear growth.

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Part Cshow ▾

Let

f(x) = x^{2}

\mathbb{R}

and

g(x) = \sqrt{x}

[0, \infty)

. Which of the following are true?

A.f is uniformly continuous on [0, 5].✓ Correct
B.f is uniformly continuous on $\mathbb{R}$ .
C.g is uniformly continuous on $[0, \infty)$ .✓ Correct
D.g is Lipschitz on [0, 1].

Solution

(1) Continuous on a compact interval, hence uniformly continuous.

(2) Fails: |f(n + 1/n) − f(n)| $= 2 + 1/n^{2}$ stays above 2 while the arguments get arbitrarily close.

(3) | $\sqrt{x} - \sqrt{y}$ | $\le \surd$ |x − y|, so $\delta = \varepsilon^{2}$ works.

(4) Fails: the difference quotient $(\sqrt{x} - 0)/(x - 0) = 1/\sqrt{x}$ blows up near 0, so no Lipschitz constant exists there — uniform continuity does not imply Lipschitz.

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2. Differentiability, mean value theorems, Taylor, L'Hôpital

Read this as a chapter →

Lec-34 Mean Value Theorems

NPTEL · Real Analysis (S. H. Kulkarni, IIT Madras)

Rolle, Lagrange and Cauchy MVT with the exact hypotheses.

Lec-36 Taylor's Theorem

NPTEL · Real Analysis (S. H. Kulkarni, IIT Madras)

Lec-36 Application of MVT, Darboux Theorem, L Hospital Rule

NPTEL · A Basic Course in Real Analysis (IIT Kharagpur)

Darboux's theorem — the reason derivatives have no jump discontinuities.

Differentiation — MVT, Darboux, Taylor and the classic spoilers free

In plain words

The mean value theorem says that at some moment your speedometer read exactly your average speed for the trip. Nearly everything examinable here is one of its hypotheses being quietly removed.

The theorems and exactly what they need

Theorem	Hypotheses	Conclusion
Rolle	f continuous on [a,b], differentiable on (a,b), f(a) = f(b)	f′(c) = 0 for some c
Lagrange MVT	continuous on [a,b], differentiable on (a,b)	f(b) − f(a) = f′(c)(b − a)
Cauchy MVT	same for f, g	(f(b)−f(a))g′(c) = (g(b)−g(a))f′(c)
Darboux	f differentiable on [a,b]	f′ has the intermediate value property (even if discontinuous)
L'Hôpital	0/0 or $\infty/\infty$ form and lim f′/g′ exists	lim f/g = lim f′/g′
Taylor (Lagrange remainder)	$f \in C^{n}, f$ ⁽ $^{n}$ ⁺ $^{1}$ ⁾ exists	remainder f⁽ $^{n}$ ⁺ $^{1}$ ⁾ $(c)(x-a)^{n}$ ⁺ $^{1}/(n+1)$ !

Consequences you can quote

f′ ≡ 0 on an interval ⇒ f constant. (Interval is essential.)
f′ > 0 ⇒ strictly increasing. Converse fails: $x^{3}$ at 0.
|f′| ≤ M ⇒ f is M-Lipschitz.
f′ bounded on (a,b) ⇒ f uniformly continuous on (a,b).
Derivatives have no jump discontinuities (Darboux). A derivative can be discontinuous, but only in the "oscillating" way.
f differentiable and f′ bounded ⇒ f of bounded variation.
If f′(x) → L as x → a⁺ then f′(a) exists and equals L (given f continuous at a).

The standard spoilers

Function	What it kills
$f(x) = x^{2} \sin(1/x), f(0) = 0$	differentiable everywhere, f′ discontinuous at 0
f(x) = x sin(1/x), f(0) = 0	continuous, not differentiable at 0
f(x) = \|x\|	Rolle's conclusion fails without differentiability at 0
f(x) = x on [0,1], f(1) = 0	Rolle fails without endpoint continuity
f(x) = x + sin x, g(x) = x	L'Hôpital for $\infty/\infty: f'/g' = 1 + \cos x$ has no limit, yet f/g → 1
f(x) = $e^{-1/x^{2}}$ , f(0) = 0	$C^\infty$ , all derivatives 0 at 0, Taylor series ≢ f
$f(x) = x^{3}$	f′(0) = 0 but strictly increasing; not a local extremum

Extrema checklist

Critical point f′ = 0 is necessary (interior), not sufficient.
Second derivative test inconclusive when $f'' = 0: x^{3} ($ no extremum $), x^{4} ($ minimum).
On a compact interval, compare critical points and endpoints.

Key takeaways

Rolle ⊂ Lagrange ⊂ Cauchy MVT; Darboux is the one that needs no continuity of f′.
L'Hôpital is one-directional: f′/g′ may fail to have a limit while f/g does. $- x^{2} \sin(1/x)$ is the universal "differentiable but not $C^{1}$ " example.

Where L'Hôpital's rule quietly failsinteractive

This visual is part of Trap Analysis

A limit that exists but which L'Hôpital never finds — and the hypothesis everyone forgets to check.

Unlock interactive visuals

Exam focus: MVT-family questions are about which hypothesis is missing: Rolle needs continuity at the endpoints, Darboux needs only differentiability, L'Hôpital needs the derivative-quotient limit to exist. Know x² sin(1/x) cold.

Practice

Part B2025 Juneshow ▾

Let

f(x) = x log_{e}(1 + 1/x)

for

x \in (0, \infty)

. Which of the following statements is true?

A.f is unbounded.
B.f is increasing.✓ Correct
C. $lim_{x\to\infty}$ f(x) = 2.
D.f is decreasing.

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Juneshow ▾

Let f be a bounded, twice continuously differentiable real-valued function on

(0, \infty)

such that f″(x) ≥ 0 for all

x \in (0, \infty)

. Which of the following statements are true?

A.f′(x) ≤ 0 for all x > 0.✓ Correct
B. $lim_{x\to\infty}$ f′(x) = 0.✓ Correct
C. $lim_{x\to\infty}$ x f′(x) need not exist.
D. $lim_{x\to\infty}$ x f′(x) = 0.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Juneshow ▾

Define

f : \mathbb{R} \to \mathbb{R}

by f(x) = x|x|. Which of the following statements are true?

A.f is continuous on $\mathbb{R}$ ✓ Correct
B.f is differentiable on $\mathbb{R}$ ✓ Correct
C.f is differentiable only at 0
D.f is not differentiable at 0

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

Consider the function

f : \mathbb{R} \to \mathbb{R}

defined by

f(x) = x^{2} \sin(1/x)

if x ≠ 0, and f(x) = 0 if x = 0. Which of the following statements are true?

A.lim(x→0) f(x) exists.✓ Correct
B.f is continuous at 0.✓ Correct
C.f is differentiable at 0.✓ Correct
D.lim(x→0) f′(x) does not exist.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2023 Decembershow ▾

Let

f : \mathbb{R} \to \mathbb{R}

be defined by

f(x) = (1 - x)^{2} \sin(x^{2})

for x ∈ (0, 1) and f(x) = 0 otherwise, and let f′ be its derivative. Let S = {

c \in \mathbb{R} : f'(x) \le c f(x)

for all

x \in \mathbb{R}

}. Which one of the following is true?

A.S = ∅✓ Correct
B.S ≠ ∅ and S is a proper subset of $(1, \infty)$
C. $(2, \infty)$ is a proper subset of S
D.S ∩ (0, 1) ≠ ∅

The worked solution and the per-option analysis are in Trap Analysis.

Part B2023 Decembershow ▾

Let f(x) be a cubic polynomial with real coefficients. Suppose that f(x) has exactly one real root and that this root is simple. Which one of the following statements holds for ALL antiderivatives F(x) of f(x)?

A.F(x) has exactly one real root.
B.F(x) has exactly four real roots.
C.F(x) has at most two real roots.✓ Correct
D.F(x) has at most one real root.

Solution

F′ = f changes sign exactly once, so F is monotone on each side of one point: at most two real roots (and the constant of integration can realise 0, 1 or 2).

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Part B2023 Decembershow ▾

Let

f : \mathbb{R} \to \mathbb{R}

be a differentiable function such that f and its derivative f′ have no common zeros in [0, 1]. Which one of the following statements is true?

A.f never vanishes in [0, 1].
B.f has at most finitely many zeros in [0, 1].✓ Correct
C.f has infinitely many zeros in [0, 1].
D.f(1/2) = 0.

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾

Let

f : \mathbb{R} \to \mathbb{R}

be differentiable everywhere. Which of the following are true?

A.f′ has the intermediate value property, even where f′ is not continuous.✓ Correct
B.f′ must be continuous.
C.If f′(x) > 0 for all x, then f is strictly increasing.✓ Correct
D.If f has a local maximum at c, then f′(c) = 0.✓ Correct

Solution

(1) is Darboux's theorem — derivatives take intermediate values whether or not they are continuous.

(2) fails for $f(x) = x^{2}\sin(1/x), f(0) = 0$ , whose derivative exists everywhere but oscillates at 0.

(3) is the mean value theorem: $f(b) - f(a) = f'(\xi)(b - a) > 0$ .

(4) is Fermat's interior-extremum theorem, applicable since f is differentiable on all of $\mathbb{R}$ .

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Part Bshow ▾

Let

f(x) = x^{2} \sin(1/x)

for x ≠ 0 and f(0) = 0. Then

A.f is differentiable everywhere and f′ is continuous
B.f is differentiable everywhere but f′ is discontinuous at 0✓ Correct
C.f is not differentiable at 0
D.f is differentiable everywhere and f′ is bounded on [−1, 1] but f′(0) ≠ 0

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾

Which of the following limits can be evaluated by a direct application of L'Hôpital's rule (i.e. lim f′/g′ exists)?

A. $lim_{x\to0}$ (sin x)/x✓ Correct
B. $lim_{x\to\infty}$ (x + sin x)/x
C. $lim_{x\to0^{+}}$ x ln x✓ Correct
D. $lim_{x\to0}$ $(1 - \cos x)/x^{2}$ ✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾

Let

f : \mathbb{R} \to \mathbb{R}

be differentiable. Which of the following are true?

A.f′ has the intermediate value property✓ Correct
B.If f′ is bounded then f is uniformly continuous✓ Correct
C.f′ is continuous
D.If f′(x) ≠ 0 for all x then f is injective✓ Correct

Solution

(a) Darboux.

(b) MVT gives Lipschitz.

(d) Darboux ⇒ f′ has constant sign ⇒ strictly monotone.

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