For real numbers a, b, c, d, e, f, consider the function F : ℝ² → ℝ² given by F(x, y) = (ax + by + c, dx + ey + f). Which of the following statements are true?

Part CCSIR NET December 2023

Partial derivatives, differentiability, chain rule

For real numbers a, b, c, d, e, f, consider the function $F : \mathbb{R}^{2} \to \mathbb{R}^{2}$ given by F(x, y) = (ax + by + c, dx + ey + f). Which of the following statements are true?

A.F is continuous.✓ Correct
B.F is uniformly continuous.✓ Correct
C.F is differentiable.✓ Correct
D.F has partial derivatives of all orders.✓ Correct

The idea, in plain words

This is just a straight-line map: it stretches, rotates and shifts, and does nothing else. Every smoothness property you could ask for holds automatically, which is why all four statements are true at once.

Solution

An affine map is Lipschitz (hence uniformly continuous), differentiable with constant derivative, and all higher partials are zero.

Why each option is right or wrong

A.Correct
True.

Each component is a polynomial of degree one in (x, y), and polynomials are continuous.
B.Correct
True.

An affine map is Lipschitz: |F(p) − F(q)| ≤ ‖A‖·|p − q| where A is the matrix of linear coefficients.

A Lipschitz map is uniformly continuous, with $\delta = \varepsilon/$ ‖A‖ working everywhere at once.

Note this holds on all of $\mathbb{R}^{2}$ — no compactness needed

which is why the usual continuous-but-not-uniformly-continuous worry does not arise.
C.Correct
True.

F is differentiable with DF(x, y) = A, the constant matrix [[a, b], [d, e]].

The remainder in the definition of the derivative is identically zero

since F is already affine.
D.Correct
True.

The first partials are the constants a, b, d, e, and every partial of order two or higher is zero.

Constants are infinitely differentiable

so F is $C^\infty$ .

All four options are correct — worth noticing

since a Part C question with no false statement is unusual and can make you doubt correct work.

The answer key is official; the reasoning above could not be confirmed by symbolic computation, so it has had no second check.

Partial derivatives, differentiability, chain rule

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