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Examples 6.4.2(a):

Let f(x) = x2. Show that f is continuous by proving
  1. that the inverse image of an open interval is open.
  2. that the inverse image of a closed interval is closed.

f(x) = x2
First, let's look at the inverse images of an open interval:

But it is now obvious that the inverse image of closed intervals is again a closed set (note that the empty set is both open and closed).

Hence, we have proved that the function f(x) = x2 is continuous, avoiding the tedious epsilon-delta proof.

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