Suppose that f is a continuous function whose domain is an interval J and f(x_0)

Question

Suppose that f is a continuous function whose domain is an interval J and f(x_0) < 0 for some number x_0 in J. Show that there are numbers a and b in J such that (need to write equation)

errrwut · Answer

$$\int\limits_{a}^{b}f(x)dx <0$$

errrwut · Answer

I am stumped and have no idea where to begin. Any tips on the direction I should take would be much appreciated.

andrewyates · Answer

I'm not sure how rigorous they want the solution to be, but I would start by noting that the function is continuous which means it must have values $a, b $ in the neighborhood of $x_0$ (nearby) such that $f(a), f(b)$ are close to $f(x_0)$. If this were not the case, the function would "jump". 

Because all values of f in this neighborhood fall between $f(a)$ and $f(b)$, they must also be negative, making the integral negative.

errrwut · Answer

I appreciate the response. Yeah, my professor is really old and didn’t really expand on that, and there’s not enough time to stop and ask questions. 
Class: intro to PDE.

andrewyates · Answer

I would look into: https://en.wikipedia.org/wiki/Continuous_function#Definition_in_terms_of_neighborhoods which you can use to make what I said earlier rigorous. Essentially, you pick a neighborhood of $f(x_0)$ ( $ N_1(f(x_0))$ ) where all values in that neighborhood are negative and use the definition of continuity to show that there exists a neighborhood of $x_0$ such that $f$ maps to $N_1$ for all $x$ belonging to that neighborhood. Then let $a, b$ be elements of that neighborhood and you get a negative integral. If you're not quite sure about the neighborhood stuff, I suppose you could use the Epsilon-Delta definition of a limit along with the definition of continuity (using limits) to prove the same thing. This is assuming your professor really wants a rigorous proof.