I'm working though (on my own; and unfortunately I don't live near a university) a text on Real Analysis: Bartle and Sherbert, 4th Ed. Chapter 5 is "Continuous Functions". In Section 5.2 ("Combinations of Continuous Functions") one of the exercises (#3, pp. 133) asks for functions f and g which are discontinuous at some point c in their shared domain, but when the functions are multiplied that function is continuous at the point c.
One of the difficulties in studying on my own is that -- for open-ended questions like this -- it is difficult to know if my own ideas are correct. I understand the answer suggested in the book, and I think I understand the question.
However, I have no clear way of determining if MY idea is (also) correct. I came up with the two functions:
f: = (e^(1/(1-x)))/(1-x)
g: = the reciprocal of f
Thus, when multiplied together, their product-function is the constant 1, and continuous at 1 (and elsewhere). But each is discontinuous at the point x = 1.
Thus, I believe this answer is also correct? Is it?
Thanks for taking the time to help me! (I'm actively looking for a local tutor, as we're a large research-base oriented community and there are at least some mathematicians working here; but no luck so far...)