analysis - Give an example of a function $ f : [a,b] → mathbb{R}$ that is continuous...

Give an example of a function $ f : [a,b] → \mathbb{R}$ that is continuous, and a sequence $(f(x_n))$ converging to $\sup(f([a,b]))$, but for which $(x_n)$ does not converge.

I am having trouble coming up with an example, especially one that converges to $\sup$. Any and all help appreciated. Thanks!