First let , then . Well clearly each is continuous. Now define , so if uniformly we will know that is continuous.
So by the M-test we have that uniformly since (geometric series with common ratio less than 1). If you don't know what the M-test is then just follow the above link.
As for the condition on it is not needed to show continuity, but if I am correct; I think that the next step would be to show that the function is nowhere differentiable. This is where the condition is most likely used.