It's increasing. Besides, it is discontinuous at and only at .
It's increasing and it is discontinuous at and only at . So, is increasing and it is discontinuous at and at and only at those points.
More generally, for each , let
Then is increasing, since it is equal to . And it is not hard to see that it is discontinuous at if and only if (this follows from the fact that the convergence of the series is uniform, by the Weierstrass -test). The reason why I told you in the comments that it should be rather than was so that the expression makes sense, that is, so that it converges, for every .
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