Types of discontinuity

1. 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.
2. 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.✓
3. C.f(x) = $e^{1/x}$ for x < 0, f(x) = $e^{1/(x+1)}$ for x ≥ 0, has a jump discontinuity at 0.✓
4. D.If $f,g:[0,1]\to \mathbb{R}$ are of bounded variation, then fg has at most countably many discontinuities.✓

• A.WrongBoundary and endpoint

  False. Check both sides, because [x] jumps at 0. For 0 < x < 1 the integer part is 0, so f ≡ 0 on the right and the right limit is 0 — which is where most students stop. For −1 ≤ x < 0 the integer part is −1, so f(x) = −sin(1/x), which oscillates without limit as x → 0⁻. A removable discontinuity needs the two-sided limit to exist; here the left limit does not. The mistake: evaluating the side on which the floor function is harmless and never checking the other.

• B.Correct

  True. As x → 0⁺, $\log x\to -\infty$, so sin(log x) oscillates through [−1, 1] forever and no limit exists. A discontinuity is removable only when the limit exists but disagrees with the assigned value; here there is nothing to remove.

• C.Correct

  True, and it is a genuine jump because both one-sided limits exist and differ. From the left, $1/x\to -\infty$ so e^(1/x) → 0 (SymPy samples e^(1/t) down to t = −0.05 and reaches 2×10⁻${}^9)$. From the right, $e^{(}1/(x+1))\to e^1=e\approx 2.718$. Left limit 0, right limit e.

• D.Correct

  True. A product of two functions of bounded variation is again of bounded variation on [0,1] (each is a difference of monotone functions, and BV is closed under products of bounded functions). A BV function has only jump discontinuities and at most countably many of them.

1 of these 1 wrong options correspond to a named reasoning error — the same errors recur across subjects.