Q: Let $f: I \subseteq \mathbb{R} \rightarrow \mathbb{R^n}$, where $I$ is an interval with some element (not empty), be a continuous function. Prove that if $f$ is Gâteaux differentiable in $a \in I$, then it is Fréchet differentiable in $a \in I$.
In my definition of the Gateaux derivative, a function is said to be differentiable in $a$ if for all $v \in \mathbb{R}^n$, the following limit exists:
$f'(a)_v = lim_{t \rightarrow 0} \dfrac{f(a + tv) - f(a)}{t}$.
To prove that the Gâteuax derivative imples on the existence of the Fréchet derivative on the given conditions, I got to the point where I only need to prove that $f'(a)_v$ is a linear application on $v$.
But i'm struggling with this prove. Some proves that i saw use the mean value theorem, but i'm not supposed to use that.
I was guessing that the prove uses something on the conexity of the image of $f$, but i don't got to far.
Any hints? Anything will be helpful.