Is every continuous function also uniformly continuous?

Question

I have a function $f:[c, a]\cup(a, b]\rightarrow \mathbb{R}$ such that on $[c, a]$ $f$ is continuous and on $(a, b]$ again f is continuous but it is given that $f(a)\neq f(a+)=lim_{h\rightarrow0} f(a+h)$ and i know that $f(a)$ and $f(a+)$ both exists and are finite real numbers

Answer 1

What about $\sin\frac{1}{x-a}$?

Answer 2

$sin (1/x)$ on$(0,1]$ satisfies the conditions but is not uniformly continuous.

Edit: what about $x^{1/2}$ over $(0,1]