# Thread: Convergent Sequences and Continuity Proof

1. ## Convergent Sequences and Continuity Proof

Don't really know how to prove this at all, I'm much appreciative of any help.

Let f : [a,b] ⇒ ℝ be a continuous function. Show that there exists a w in [a,b] such that

a
∫ f(t)dt = f(w)(b-a)
b

Here is the hint that i need to use to solve this problem:
f attains a smallest value A and a largest value B. Show that:

-----------a
A ≤ 1/(b-a)∫ f(t)dt ≤ B
-----------b

Thank you again for the help!

2. So we have $\displaystyle A\leq\frac{1}{b-a}\int_a^bf(t)dt\leq B$.
Let $\alpha=\frac{1}{b-a}\int_a^bf(t)dt$. Then $\alpha$ is a mean value of f.
f is continuous on [a,b], then f has the Darboux property. So, exists $w\in[a,b]$ such as $f(w)=\alpha$.