Let X be a discrete space, i.e.: every subset of X is open, and let
now f: X --> H be any function from X to any top. space H.
Let U be open in H: is f^(-1)(U) open in X? Of course it is...can you
see why?

If now Y is an indiscrete top. space, then its only open subsets are
the empty set and Y itself. Any function f: Z --> Y from any top.
space Z will have only two open sets in Y to check on: so, what are
the inverse images of the empty set and of the whole space Y?

f:X -> Y is continuous when
        for all open U subset Y, f^-1(U) open subset X.

This is the definition that requires study and to assure yourself
that in encompasses the definition of continuous of metric spaces
and the usual definition of continuous in analysis.

Another definition with the same need for study.
f:X -> Y is continuous at a when for all open V nhood f(a),
some open U nhood x with f(U) subset V.

Theorem.  f:X -> Y is continuous iff for all a in X,
        f is continuous at a.

Anyway, let X be a discrete space.
If U open subset Y, then of course f^-1(U) is open.
Thus immediately f is continuous.