§ Pasting lemma
Let be a function.
Let are closed subets of such that . Then is continuous iff
and are continous.
§ Forward: continuous implies restriction continuous
This is clear from how restrictions work. Pick to be the
function that embeds with the subspace topology into . This is continuous by
the definition of the subspace topology. Now define .
which is continous since it is the composition of continuous functions.
§ Backward: restrictions continuous implies continuous.
Let be closed. Then is closed in by the continuity of
Now see that is closed in the subspace topology of means that
there is some closed such that .
Since both and are closed in , this means that is closed
in (see that we have filted "closed in " to "closed in ).
Similarly, we will have that for some closed and .
Then we can write: