question about uniform continuity I have a function $f:[a, b]\rightarrow \mathbb{R}^n$ such that $f$ is piecewise continuous in the intervals $[a, t_1],$ $(t_1, t_2],$ $(t_2, t_3],$...., $(t_n, b].$ Can we say that $f$ is uniformly piecewise continuous on $[a, b]?$

No. A counter example is $f: [-1,1] \to \mathbb R$ defined as
$$f(x)=\begin{cases}0, &-1 \leq x \leq 0 \\ \frac 1x, &0<x\leq 1 \end{cases}$$
