Is convergence of elementary (simple) functions always uniform?

Question

Say I have a bounded continuous function $g(t)$ on the interval $[a,b]$ Let $$g_{n}(t)=\sum_{j=1}^{n}g(t_{j})X_{[t_{j},t_{j+1}]}(t)$$

Where X is the characteristic function, and $t^{n}_{j}$ a partition of $[a,b]$ where $max_{j=1...n}|t_{j+1}-t_{j}|\to 0$ as $n\to \infty$

Then is the convergence of $g_{n}$ to $g$ uniform?

Answer 1

For each $n$ we have a partition: the $t_j$ depend on $n$. They should be $t_j^n$, $1\le j\le n+1$. Assuming $\max(t_{j+1}^n-t_j^n)\to0$, the answer is yes, because $g$ is uniformly continuous.