# The uniform continuity of functions with Banach space values. [duplicate]

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 $$[a,b]$$?