Proving Integrability: f on [a,b] with c ∈ [a,b]

[dgonnella89]

Homework Statement

Prove if f is integrable on [a,b] then $$\exists c$$ $$a\leq c \leq b$$ so that $$\int_a^cf = \int_c^bf$$

The Attempt at a Solution

I know that you have to use the first fundamental theorem of calculus and start with:
$$g(x) = \int_a^xf(t)dt-\int_x^bf(t)dt$$

I'm not sure how to get started and to use this fact to help prove it. Any help would be really appreciated. Thanks!

 

[Dick]

g(x) is a continuous function, right? Use the intermediate value theorem. What are g(a) and g(b)?