In [this article](http://www.jstor.org/stable/2319406) (DOI: 10.2307/2319406 ), the author define a function as 

$$h^{+}(x) = \max\{h(x), 0\},$$
where $h: \mathbb{R}^n \to \mathbb{R}$ and $h \in C^1 (G), G\subseteq \mathbb{R}^n$.

**Is it the case that $h^{+}$ is always continuously differentiable ?** 

My gut says that it should not necessarily be because at the points where $h(x) = 0$, $h^+$ might be not continuously differentiable because of that steep change fro the constant value $0$ to $h(x) > 0$.

Edit:

In case you cannot access to the article: This is the section that the claim is made.

[![enter image description here](https://i.stack.imgur.com/BbvDE.png)](https://i.stack.imgur.com/BbvDE.png)