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?
Thanks!
What I wrote in my OP is the original wording. I assume that since this is a textbook on real analysis, we have and that and , where , but perhaps my assumptions are incomplete (or even wrong).
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.