Checking Continuity and Differentiability

Question

Let $f(x)=\sin|x| ,x \in(-\pi ,\pi)$

Is $f$ continuous in the interval $(-\pi ,\pi)$ ? If it is, then is it differentiable on $(-\pi ,\pi)$ ??

Is it a problem of uniform continuity ? How to solve it?

Answer 1

The function $f$ is continuous since $\sin y$ is continuous and $y = |x|$ is continuous. The same argument shows that it is differentiable at least for all $x\neq 0$. Check the case $f'(0)$ by yourself using the definition of the derivative directly.