Give an example (or prove that none exists) of a real function f(x)
which is continuous, invertible, and satisfies the following identity everywhere on
its domain of definition: f^-1(x)=1/f(x)

Thank you for the opportunity to help you with your question!

For functions, the notations
mean the same thing, but "f(x)" provides more flexibility and more
information. You used to say "y=4x+ 3; solve for y
when x = –1". Now you say f(x)= 4x + 3; find f(–1)" (pronounced as "f-of-x
is 4x plus three; find f of negative one.

Do exactly
the same thing for each: you plug in –1 for x.
and *2, and then +3, simplifying to
get a value of -1

In the question, f(x) will be contionous when the values of f^-1(x) will be equal to that of its reciprical.

Please let me know if you need any clarification. I'm always happy to answer your questions.