real analysis - Intermediate Value Theorem and Discontinuous Functions

I am asked to find an example of a discontinuous function$ f : [0, 1] → \mathbb{R}$ where the intermediate value theorem fails. I went over the intermediate value theorem today

Let $f : [a, b] → \mathbb{R}$ be a continuous function. Suppose that there exists a $y$ such that $f(a) < y < f(b) $ or $ f(a) > y > f(b).$ Then there exists a$ \ \ c ∈ [a,b]$ such that $f(c) = y$.

I understand the theory behind it, however, we did not go over many example of how to use it to solve such problems so I do not really know where to begin