# Piece-wise function Differentiable?

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 $$x=0$$.