Answer
True.
(See the step-by-step explanation for proof.)
Work Step by Step
Let us assume the opposite. If we arrive at a contradiction,
we have to abandon our assumption and accept that the given statement is true.
Given: f is continuous, never zero.
Assumption: f changes sign at least once on the interval.
Then, there are numbers a and b on the interval such that
f(a) and f(b) have opposite signs.
$y_{0}=0$ is a value between f(a) and f(b).
The Intermediate Value theorem guarantees that there exists a $c\in(a,b)$ for which $f(c)=0$.
This is a contradiction with "f is never zero."
Thus, our assumption was wrong -- f NEVER changes sign on the interval.
