Limit of composition of non-continuous functions

Assume ff and gg are defined on all of R\mathbb{R} and limx→pf(x)=qlim_{x\to p} f(x) = q and limx→qg(x)=rlim_{x\to q} g(x) = r. Give an example to show that it may not be true that limx→pg(f(x))=rlim_{x\to p} g(f(x)) = r

I’ve toyed with various ideas, including Dirichlet’s and Thomae’s functions, but I can’t seem to come up with an example.

=================

=================

1 Answer
1

=================

Hint: Take f(x)f(x) to be constant, and remember that limx→qg(x)\lim_{x\to q}g(x) doesn’t depend on g(q)g(q).