Single Variable Calculus, Chapter 2, 2.5, Section 2.5, Problem 44

If a function $f$ is continuous on $[0, 1]$ except at $0.25$ and that $f(0) = 1$ and $f(1) = 3$. Let $N = 2$. Sketch two possible graphs of $f$, one showing that $f$ might not satisfy the conlusion of the Intermediate Value Theorem and one showing that $f$ might still satisfy the conclusion of the Intermediate Value Theorem.







The first graph does not satisfy the Intermediate Value Theorem since the function is discontinuous at the given interval $[0, 1]$ and $N= 2$ does not intersect the given function at any point.



Suppose that we add another function $f(0.25) = 2$ as shown in the graph below.








Therefore, the function is now continuous on the interval (0,1) and satisfies the Intermediate Value Theorem.