My real analysis textbook asks me to
Let . Show that .
It looks like a fairly simple problem, at the end of the chapter on the chain rule. However, when I apply the chain rule, I get
Now, if I can show that either or is continuous on , then I can let and the conclusion follows. But how do I prove that with so little information? Or am I going about this entirely the wrong way?
If you doubt me, the textbook is freely available from the author in pdf format, here.
See exercise 12a from section 5.4, p358---or pdf page 367.