It is clear that the composition of two $C^i$ function is still a $C^i$. But my question is more about a kind of its reciprocal.
Let's consider two open interval $I_1$ and $I_2$ in $\mathbb{R}$.
Then we have a function $g: I_1 \rightarrow I_2$ that is $C^{i+1}$, bijective and his derivate is non-zero in a point $x_0$ and $f: I_2 \rightarrow \mathbb{R}$ that is $C^i$ but not $C^{i+1}$ in $y_0:=g(x_0) \in I_2$.
My question is, can the composition function $f \circ g$ be $C^{i+1}$? Or can a find a function $g$ that "erase" the $C^{i+1}$-discontinuity for every such fuction $f$?