This question already has an answer here:

Let $f\colon [a,b]\to\mathbb{R}$ be a continuous function. Since $[a,b]$ is compact, then by continuity of $f$ we also have that $f$ is uniformly continuous on $[a,b]$. Suppose now that $F\colon [a,b]\to X$ is continuous. In here, $X$ is a Banach space. For sure we have that the map $[a,b]\ni x\mapsto\|F(x)\|$ is uniformly continuous on $[a,b]$. But, do we have that $F$ is uniformly continuous on $[a,b]$?


marked as duplicate by freakish, Community Mar 19 at 11:29

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.