Function f should be _____ on [a,b] according to Rolle’s theorem

The correct option is continuous.

Rolle's theorem is a fundamental theorem in calculus that describes the behavior of a differentiable function on a closed interval. For the theorem to apply, the function must satisfy specific conditions.

Specifically, Rolle's theorem states that if a real-valued function $f$ is:

1. Continuous on the closed interval $\left[a,b\right]$,
2. Differentiable on the open interval $\left(a,b\right)$, and
3. $f\left(a\right)=f\left(b\right)$,

then there exists at least one number $c$ in the open interval $\left(a,b\right)$ such that the derivative $f^{\text{'}}\left(c\right)=0$.

Therefore, the function $f$ must be continuous on the closed interval $\left[a,b\right]$ as the first essential hypothesis of the theorem.