Being able to help you is dependent on what you know.

Do you know that if a functionfis continuous on a compact set thenfis uniformly continuous there? Thus if then interval is compact (closed and bounded) sofis uniformly continuous there. Also we are given thatfis uniformly continuous on , but

The part (b) is just a bit more difficult.

Say that and .

From above you can see thatfis uniformly continuous on

Now prove thatfis uniformly continuous on .