Show that $f$ is uniformly continuous given $\lim_{x \to -\infty}f(x)=a$ and $\lim_{x \to \infty}f(x)=b$

Question

Let $f:\Bbb R \to \Bbb R$ be continuous. Assume $$ \begin{aligned} &f(x) \rightarrow a \quad \text { for } x \rightarrow-\infty\\ &f(x) \rightarrow b \quad \text { for } x \rightarrow \infty \end{aligned} $$ Show that $f$ is uniformly continuous.

I'm lost here.

Answer 1

Hints:

For "very small" $x_1, x_2$, you have $|f(x_1)-f(x_2)|=|f(x_1)-a+a-f(x_2)|\leq |f(x_1)-a|+|f(x_2)-a|$.

For "very large" $x_1, x_2$, you have $|f(x_1)-f(x_2)|=|f(x_1)-b+b-f(x_2)|\leq |f(x_1)-b|+|f(x_2)-b|$.

For everything in between, functions, continuous on closed intervals, are uniformly continuous.