I have a question about uniform convergences of functions in $\mathbb{R}^d$. In $\mathbb{R}$, we know that if $\{f_n\}_{n \in \mathbb{N}} \subset \mathbb{R}$, $f'_n$ converges uniformly towards $h$ and it exists $x_0 \in \mathbb{R}$ such that $f_n(x_0)$ converges, then $f_n$ converges uniformly.
But now, my question is : in $\mathbb{R}^d$, if we have the uniform convergence of the partial derivatives, i.e $\forall i \in \{1, ..., d\} \quad (\frac{\partial f_n}{\partial x_i})_n$ converges uniformly, and if it exists $x_0 \in \mathbb{R}^d$ s.t $f_n(x_0)$ converges, then can we conclude that $f_n$ converges uniformly ?
Thank you !
The result you say we know is false. First, $\{f_n\}\subset\Bbb R$ is not what you meant. That says $(f_n)$ is a sequence of real numbers, not a sequence of functions. You meant to say $f_n:\Bbb R\to\Bbb R$.
Let $f_n(t)=t/n$, $h(t)=0$, $x_0=0$. Then $f_n'\to h$ uniformly, $f(x_0)$ converges, but $(f_n)$ is not uniformly convergent.
I could tell you what the correct version is; better would be for you to look up the result wherever you learned it from and get it straight yourself.