The statement
If $f$ is continuous on $[a,b]$ then $f(a)$ and $f(b)$ aren’t necessarily local extrema. Though they are extrema $f$ is strictly monotonous to the right of a and the left of b respectively.
I understand why this is the case and it can be shown by the piecewise function $$f(x)=\begin{cases}x\sin\frac{1}{x} ,\text{ if } x>0 \\ 0 , \text{ if }x=0 \end{cases}$$ But I do feel that a more rigorous statement is suited here though I understand why it isn't since it's from a high school book.
My attempt
From the little I have seen from higher level math this statement is better expressed with deltas, similar to the definition of local extrema.
If $f$ is continuous on $[a,b]$ then $f(a),f(b)$ are local extrema iff $\hspace{0.3em}\exists\delta>0$ s.t. $f$ is strictly monotonous on $[a,\delta)$ and $(\delta,b]$ respectively.
I know this isn't what a mathematician or anyone who took an analysis course would call rigour since I didn't even rigorously define the continuity on $[a,b]$. It's an attempt from a high school student for the sake of understanding what is being stated.
Note that $$f(x)=\begin{cases}2x+x\sin\frac1x &x>0\\0&x=0\end{cases}$$ is continuous on $[0,\infty)$ and has a (global) minimum at $x=0$. However, there is no $\delta>0$ such that $f$ is monotonous on $[0,\delta)$.