Consider the following function $f : \Bbb R \to \Bbb R$:

$$f(x) = \begin{cases} \sin x & x \ge 0 \\ x^2 + e^x & x < 0 \end{cases}$$

I was wondering if the above piece-wise function was differentiable over the set of Real numbers R, and how to figure it out.

I know that differentiability implies continuity, so maybe I could check if it was continuous as a starter?


That is a good idea. Now check the $x=0$ point. For the right side of the graph, $\sin 0 =0$. For the left side, $0^2+e^0=1$. It is not continuous at $x=0$, and is therefore not differentiable at $x=0$.


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