Is a function with differentiable parts and non-differentiable parts differentiable?

Question

I was in Computer Vision + Deep Learning Lecture, and talking about some Loss functions and other elements, and they mentioned that in a particular function, by applying the chain rule, some parts of the derivative of the function are not differentiable, so something must be done with that.

From that, I started thinking whether a function that has some differentiable parts and some non-differentiable parts is differentiable. My direct answer is: NO, it isn't differentiable. But right now I cannot think of a demonstration for it. So, any available demonstration about it?

For example,

$$ f(x) = x + |x|$$

Has clearly a differentiable part ($x$) and a non differentiable part $|x|$.

Answer 1

Let $D$ be a non-void subset or $\mathbb R$ and $f,g,h:D \to \mathbb R$ functions with the following properties:

1. $f=g+h$,

2. $g$ is differentiable on $D$ and $h$ is not differentiable on $D$.

Now suppose that $f$ is differntiable on $D$. Since $h=f-g$, it would follow that $h$ is differentiable on $D$, a contradiction.