Sum of differentiable functions. True/False Question : suppose that $f+g$ is differentiable at point $x_0$ therefore $f$ and $g$ are differentiable at $x_0$ I think this statement is false and I got a counter example : $f(x)=\begin{cases} 1 & \,\,x>0\\\ x & \,\,x<0 \end{cases}; g(x)=\begin{cases} x\,\, & x-1>0\\\ 0 & \leq0 \end{cases}$ $(f+g)(x)=\begin{cases} x\,\, & x>0\\\ x\,\, & x<0\\\ 0 & x=0 \end{cases}\rightarrow\forall x\in\mathbb{R}\,\,\left(f+g\right)(x)=x, \left(f+g\right)'(0)=0$ Nor $f$ or $g$ are differentiable at $x_0=0$ but the sum is differentiable. **However second True/false question got me very confused, it says** : suppose that $f+g$ and $f$ is differentiable at point $x_0$ therefore $g$ are differentiable at $x_0$ and here I got confused, I can not find acounter example however I can not also find a proof

Use the fact that $(f+g)-f=g$ and that the difference of differentiable functions is differentiable.