Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ be a function such that both $f$ and $\frac{\partial^2 f}{\partial x^2}$ are continuous on $\mathbb{R}^2$. Is then $\frac{\partial f}{\partial x}$ also continuous on $\mathbb{R}^2$?
Precise statement: Let$$g:\mathbb{R}^2\rightarrow\mathbb{R}:(x,y)\mapsto\lim_{\varepsilon\rightarrow0}\frac{f(x+\varepsilon,y)-f(x,y)}{\varepsilon},$$and let:$$h:\mathbb{R}^2\rightarrow\mathbb{R}:(x,y)\mapsto\lim_{\varepsilon\rightarrow0}\frac{g(x+\varepsilon,y)-g(x,y)}{\varepsilon},$$then, assuming $g$ and $h$ both exist, if $f$ and $h$ are continuous, is $g$ also continuous?
This looks quite elementary but unusual as we don't deal with any $C^k$ function. I am quite lost with this one. Any help?