Let $\{f_v\}_v\in\mathbb N$ be a sequence of continuous functions $f_v:\Re^m\to\Re^n$ and $f:\Re^m\to\Re^n$ and assume that $f_v$ converges to $f$ pointwise (i.e. For every fixed $x\in\Re^m$, $f_v(x)\to f(x)$). If the sequence $f_v$ was equicontinuous, then it would also converge uniformly to $f$. I am looking for some alternative conditions that may render $f_v$ also uniformly convergent to $f$.
Consider the continuous functions $f_v(x)=x^v$, where $x\in[0,1]$. These functions converge pointwise to $f(x)=1-\chi_{[0,1)}$ but they do not converge uniformly (and of course they are not equicontinuous). I notice additionally that their pointwise limit is not continuous.
Consider now the following alternative conditions:
- $f_v:\Re^m\to\Re^n$ converges point-wise to $f$
- $f_v$ are continuous
- The sequence $f_v$ is uniformly bounded, i.e. there is a compact set $K\subset \Re^n$ such that $f_v(x)\in K$ for all $v\in \mathbb N$ and for all $x\in\Re^m$.
- $f$ is continuous!
Under these assumption, can we show the following property for the sequence $f_v$ (referred to as continuous convergence):
Fix a $x_0\in\Re^m$. For every sequence $x_v\to x_0$, it holds that $f_v(x_v)\to f(x)$.
Can we also show that the sequence $f_v$ converges uniformly to $f$?
Update: After the comment of a user, I thought of adding one additional condition:
- The sequence $f_v$ is uniformly Lipschitz at $0$, i.e. there is an $L>0$ such that $\|f_v(x)\|\leq L\cdot\|x\|$ for all $x\in \Re^m$ and for all $v\in\mathbb N$.