uniform continuity over $(0,1)$

Question

given a function $ f(x)= e^{x^2}\sin(x^2) $ I have to check whether it is uniformly continuous over the interval (0,1). From past experience I know how to deal with the problem if $ f(x)= \sin(x^{2}) $ in which we take two subsequences say $ x_{n}=\sqrt{2n\pi}$ and $y_{n}=\sqrt{(n+\frac {1}{2})\pi} $

But for this problem I am clueless ! so please provide some hints so that I can proceed further !

Answer 1

Any continuous function on the compact space $[0,1]$ is uniformly continuous and the restriction to $(0,1)$ of a such a function is automatically uniformly continuous.