Uniform convergence
Let be $X$ a topological space and $\{f_n\}_{n\in\mathbb{N}}$ a sequence of functions from $X$ to $I\subseteq\mathbb{R}$. So we say that the sequence $\{f_n\}_{n\in\mathbb{N}}$ is uniformly convergent to a real-valued function $f$ if for every $\epsilon>0$ there exist a $m$ such that we have $$ |f(x)-f_n(x)|<\epsilon $$ for every $x\in X$ and $n\ge m$.
So I know that if $f$ is a scalar function of class $C^1$ defined in a subset $S\times I$ of $\Bbb R^n\times\Bbb R$ then $$ \lim_{n\rightarrow+\infty}\frac{f(x,t_n)-f(x,t_0)}{t_n-t_0}=\frac{\partial f}{\partial t}(x,t_0) $$ for each $x\in S$ when $t_n$ converges to $t_0$. So that the sequence $$ \partial f_n(x):=\frac{f(x,t_n)-f(x,t_0)}{t_n-t_0} $$ point wise converges to $\partial_{t_0}f(x):=\frac{\partial f}{\partial t}(x,t_0)$ but unfortunately I did not able to understand if it converges uniformly too. So could someone show to me if $\partial f_n$ converges to $\partial_{t_0}f$ uniformly, please? In particular if the statement is true then how prove it?