I do not understand why the writer said:"If $f$ is continuous everywhere, then it is uniformly continuous" in this paragraph:
Could anyone explain this for me please?
The detailed theorem and its prove is found here: Difficulty(2) in understanding Thm 4.1 in Stein & Shakarachi Fourier Analysis
You are not providing context (or rather, you want us to chase it down through two other questions). The domain of $f$ is the circle, which is compact. A continuous function on a compact set is uniformly continuous.