Prove: If a continuous function is monotonic in $(a, b) $ then it is monotonic in $[a, b] $

Question

If a continuous function is monotonic in $(a, b) $ then it is monotonic in $[a, b] $

I conjectured this theorem for this would solve many doubts I had in monotonicity of functions. I can see this intuitively but can someone provide a bit of rigour?

For example, this can be used to prove that $y=x^3$ is monotonic for $x=0$.

Answer 1

Hint: Use the Intermediate Value Theorem.