Question

Let f(x) = |sin x| Then

• A. f is everywhere differentiable
• B. f is everywhere continuous but not differentiable at x = nπ, n ∈ Z
• C. f is everywhere continuous but no differentiable at x = (2n + 1) π/2 n ∈ Z
• D. None of these

Answer 1

Answer: (B)

f is everywhere continuous but not differentiable at x = nπ, n ∈ Z

Answer 2

The function $f(x) = |\sin x|$ is a composition of $\sin x$ and $|x|$. Both are continuous functions, so their composition is continuous everywhere. The non-differentiability of $|g(x)|$ occurs where $g(x) = 0$ and $g'(x) \neq 0$. For $f(x) = |\sin x|$, $\sin x = 0$ at $x = n\pi$ for $n \in \mathbb{Z}$. At these points, $\frac{d}{dx}(\sin x) = \cos x = (-1)^n \neq 0$. Thus, $f(x)$ is not differentiable at $x = n\pi$. At other points, $f(x)$ is locally either $\sin x$ or $-\sin x$, both of which are differentiable.