Prove: If a continuous function is monotonic in $(a, b) $ then it is monotonic in $[a, b] $ > If a continuous function is monotonic in $(a, b) $ then it is monotonic in $[a, b] $ I conjectured this theorem for this w...--prophetes.ai

Prove: If a continuous function is monotonic in $(a, b) $ then it is monotonic in $[a, b] $ > 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 $x=0$.

**Hint:** Use the Intermediate Value Theorem.