# Convexity of Inverse Function if Not Differentiable

Suppose you have a $f:\mathbb{R} \rightarrow \mathbb{R}$ that is convex. Can you say anything about the convexity of its inverse when the function is not differentiable? If the function is differentiable, then f convex means that $f''(x)>0$. In this case, I think you can say something about the convexity of the inverse by taking its second derivative, since the first derivative of the inverse is $\frac{d}{dx} f^{-1}(x)=\frac{1}{f'f^{-1}(x))}$ But what if the function is not differentiable? Can you still make any conclusions about the convexity of its inverse?