If $f$ is a continuous function on $[0,2]$ and $f(1) = -1$ , $f(2)=1$ then there must be a point $x$ in $[0,1]$ where $f(x) = 0$

Question

If $f$ is a continuous function on $[0,2]$ and $f(1) = -1$ , $f(2)=1$ then there must be a point $x$ in $[0,1]$ where $f(x) = 0$.

Is it true or false? Can you justify if true?

Answer 1

$f(1)=-1 \leq 0 \leq f(2)=1 $, and $f$ continuous , by the intermediate values theorem since there is $x \in [0,2]$ such that $f(x)=0$

Answer 2

It is not true . The intermediate value theorem guarantees $f(x)=0$ for $x\in [0,2]$ but $f(x)=0$ might no happen in [$0,1]$.