if $f$ is continuous, then $f$ is uniformly continuous

Question

If every continuous funtion $f:X\to \mathbb{R}$ is uniformly continuous, where $X \subset \mathbb{R}$. Then $X$ is closed. But not necessarily compact

It's like the reciprocal of continuous $f$ and $X$ compact, then $f$ is uniformly continuous. But I do not know where to start

help

Answer 1

If $X$ is not closed then there exists a limit point $x_0 \in \mathbb{R}$ not contained in $X$. Then, the function

$$f(x) := \frac{1}{x-x_0}$$

is continuous on $X$ but not uniformly continuous. Note that since this argument does not rely on $X$ being bounded or not, we cannot make any deduction about compactness of $X$.

Answer 2

If $X$ were not closed, so that $a\in \mathbb{R} \backslash X$ is a limit point for $X$, the function $f(x)=\frac{1}{x-a}$ defined on $X$ is continuous but not uniformly continuous, since it is unbounded on any neighborhood of $a$.