Answer
The statement is true.
Work Step by Step
Take a continuous function $f(x)$, which is never $0$ on the interval $[a,b]$.
The Intermediate Value Theorem states a function $f(x)$ is continuous on $[a,b]$ and if $y_0$ is between $f(a)$ and $f(b)$, there exists a value of $x=c\in[a,b]$ such that $f(c)=y_0$.
So because the continuous function $f(x)$ is never $0$ on $[a,b]$, $0$ is not between $f(a)$ and $f(b)$. Both of them are either greater than $0$ or less than $0$ and all of the values between $f(a)$ and $f(b)$ are the same.
Therefore, $f(x)$ never changes sign on $[a,b]$. The given statement is true.
