Why ReLU function is not differentiable at 0? I'm kind of rusty in calculus. 
Why is the ReLU function not differentiable at $f(0)$?
$$
f(x) = 
\begin{cases}
0 & \text{if $x \leq 0$} \\ 
x & \text{if $x > 0$}.
\end{cases}
$$

If you look at $x > 0$, or the righthand derivative, $$\frac{df}{dx} = \frac{d}{dx} x = 1$$ for all $x$.
If you look at $x \le 0$, or the lefthand derivative, $$\frac{df}{dx} = \frac{d}{dx} 0 = 0$$ for all $x$.
Since $x = 0$ is the "break" point, the lefthand and righthand derivatives are not the same, and thus, the derivative is not defined at $x = 0$.
