Prove that $f$ is uniformly continuous on $[a,b)$ if $\displaystyle\lim_{x\to b}f(x)$ exists.

Question

We have $a,b\in \mathbb R$ and $a<b$ ,also $f:[a,b)\to\mathbb R$ is continuous ,then how can we prove that $f$ is uniformly continuous if $\displaystyle\lim_{x\to b}f(x)$ exists.

My try:We know that a continuous function on closed bounded interval ,is uniformly continuous but here,the interval is not closed.So,I don't know how to solve.Thank you.

Answer 1

You have to check continuity of $f$ at $x=b$, that is show that $\lim_{x\to b} f(x)=f(b)$. This will imply that $f$ is continuous on $[a,b]$ which is a compact subset of $\mathbb{R}$ and $f$ will hence be uniformly continuous.

Answer 2

Define a new function $g:[a,b]\rightarrow \mathbb{R}$ as follows. Define $g(x)=f(x)$ when $a\le x<b$. Define $g(b)=\lim_{x\rightarrow b} g(x)$. Check that $g$ is continuous and hence uniformly continuous.