Consider the following statements, where [x] denotes the greatest integer less than or equal to x:
(A) \(f(x) = \sin|x|\) is continuous at every real number x.
(B) \(f(x) = [x]\) is continuous at \(x = 2\).
(C) \(f(x) = e^{|x|}\) is differentiable at \(x = 0\).
(D) \(f(x) = x|x|\) is differentiable at \(x = 0\).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) and (D) only
sin|x| is a composition of continuous functions (|x| and sin x), so it is continuous everywhere.
For [x] at x = 2:
\( \displaystyle \lim_{x \to 2^-}[x] = 1, \quad \lim_{x \to 2^+}[x] = 2 \)
For \(e^{|x|}\) at x = 0, the one-sided derivatives are
\( \displaystyle \lim_{h \to 0^+}\frac{e^{h} - 1}{h} = 1, \quad \lim_{h \to 0^-}\frac{e^{-h} - 1}{h} = -1 \)
For x|x| at x = 0:
\( \displaystyle \frac{f(h) - f(0)}{h} = \frac{h|h|}{h} = |h| \to 0 \)
So sin|x| is continuous everywhere, [x] jumps at 2, \(e^{|x|}\) has one-sided derivatives −1 and 1 at 0, and x|x| has derivative 0 at 0.
Checking the statements
(A) \(f(x) = \sin|x|\) is continuous at every real number x. — Correct. It is sin x applied to the continuous function |x|.
(B) \(f(x) = [x]\) is continuous at \(x = 2\). — Incorrect. The left limit is 1 but the right limit is 2.
(C) \(f(x) = e^{|x|}\) is differentiable at \(x = 0\). — Incorrect. The left derivative is −1 and the right derivative is 1.
(D) \(f(x) = x|x|\) is differentiable at \(x = 0\). — Correct. The difference quotient is |h|, which tends to 0, so f′(0) = 0.