If $f(x)>g(x)$ after intersection it remains that way until next intersection.

Question

I want to know how to prove that if you have 2 functions $f(x)$ and $g(x)$. And if $f(x) > g (x)$ after an intersection then it remains that way until the next intersection.

Note: $f$ and $g$ are continuous functions.

Answer 1

If two continuous real-valued functions $f$ and $g$ do not intersect on the interval $(a, b)$ then the difference $f-g$ has the same sign on the interval.

That is an immediate consequence of the “intermediate value theorem”: If $f(x_1) - g(x_1) > 0$ and $f(x_2) - g(x_2) < 0$ then $f(x) -g(x) = 0$ for some $x$ between $x_1$ and $x_2$, contrary to the assumption.

Answer 2

Let $$h(x)=f(x)-g(x)$$

Note that $h(x) $ is continuous

The intermediate theorem state that if there is a change in sign of a continuous function $h(x)$ there must be at least one $c$ such that $h(c) =0$

That provides a point of intersection for $f$ and $g$