Two things to notice. may be continuous extended to in the usual (and unique) way. Thus, for any is continuous on and thus by the Heine-Cantor Theorem we see that is unif. cont. on . Thus, any restriction of on is unif. cont., in particular is unif. cont. on . But, ... so try working with that.

If you can use something besides the strict definition and consequences I would note that is unif. cont. on as proven above and is Lipschitz on since it has bounded derivative. From there you can conclude.