Second part of Theorem 5.7 : If a function is one-to-one(and therefore has an inverse function), then must the function be strictly monotonic.

This is not true.

Counter example :

Consider the function as .

The function is one-to-one, but not monotonic.

Draw a coordinate plane.

Graph the function .

Graph :

Observe the above graph :

The local maximum is  and local minimum is .

From the graph, the statement " If a function is one-to-one and continuous, then it isstrictly monotonic ".

No, converse of the second part of Theorem 5.7 is not true.