Let $U\subseteq\mathbb R$, $f:U\to\mathbb R$ and $x\in U^\circ$. Is it possible that $f$ is differentiable at $x$ in each direction$^1$, but not Gâteaux/Fréchet differentiable?
Clearly, if $f$ is differentiable at $x$ in the "ordinary" sense, then $f$ is Fréchet differentiable at $x$. The derivative ${\rm D}f(x)$ is identified with the ordinary derivative $f'(x)$ (as it is the case for any real-valued $f$ defined on a subset of a Hilbert space by Riesz' representation theorem).
$^1$ Let $E_i$ be a normed $\mathbb R$-vector space, $U\subseteq E_1$, $f:U\to E_1$ and $x\in U^\circ$. Since the terminology is not consistent in the literature, let me briefly define the following: $f$ is called
- differentiable at $x$ in direction $h\in E_1$ if $$\frac{f(x+th)-f(x)}t\xrightarrow{t\to0}{\rm D}_hf(x)\tag1$$ for some ${\rm D}_hf(x)\in E_2$;
- Gâteaux differentble at $x$ if $f$ is differentiable at $x$ in each direction and $${\rm D}_hf(x)={\rm d}f(x)h\;\;\;\text{for all }h\in E_1$$ for some ${\rm d}f(x)\in\mathfrak L(E_1,E_2)$;
- Fréchet differentiable at $x$ if $$\frac{\left\|f(x+h)-f(x)-{\rm D}f(x)h\right\|_{E_2}}{\left\|h\right\|_{E_1}}\xrightarrow{h\to0}\tag2$$ for some ${\rm D}f(x)\in\mathfrak L(E_1,E_2)$.