How to show that f is continuous over interval?

Question

Let f be continuous and never zero on [a,b]. Show that either f(x)>0 for all x in [a,b] or f(x)<0 for all x in [a,b]

Answer 1

Hi Gnarls B.

Logically, if the function is continuous and never zero, it must stay to one side of the other! I suppose you could formalize that by interpolating the 3 possible behaviors (stays below, stays above, or crosses one or more times) and then discarding the third behavior as violating the stated conditions. You might also need to work up a formal proof of "behavior" over infinitesimal and finite intervals, I suppose. Something with mean value theorem, or some such.

-- Cheers, -- Mr. d.