How to prove using epsilon-delta definitions that a function is continuous over a smaller interval

Question

I'm struggling to use the epsilon-delta definitions of continuity to show that if $f(x)$ is continuous over the interval $(a,b)$ then is is also uniformly continuous of the interval $[a+1,b-1]$ and hence also uniformly continuous over the interval $(a+1,b-1)$.

If anyone could help that would be great cheers.