Proof that a continuous function on $[a,b)$ is bounded (epsilon-delta)

Question

Prove that if $f$ is continuous on $[a,b)$ and $\lim_{x\rightarrow b^-} f(x)$ exists, then $f$ is bounded, and for every $\varepsilon > 0$ there is a $\delta > 0$ such that $|f(x)-f(y)| < \varepsilon$ for every two numbers $x, y \in [a, b)$ with $|x-y| < \delta$.

Answer 1

If that happens, we can extend $f$ continuously in the obvious manner to $[a,b]$. All your results now follow from the fact that the extension of $f$ is continuous in a compact set (Your results: namely, uniform continuity and boundedness).