Convergence? (Calculus)

[beekerz]

Verify that if f(x) is continuous on [a,b], then
\(\displaystyle |\int_a^b \! f(x) \, dx| < \infty\).

I'm not really sure how to answer this because it just seems like it's an easy question. Is it because it converges at b or from b to infinity?

 

[pka]

Verify that if f(x) is continuous on [a,b], then
\(\displaystyle |\int_a^b \! f(x) \, dx| < \infty\).

The is a very important theorem about continuous functions on [a,b].
If f(x) is continuous on [a,b] then there at two points \(\displaystyle \{c,d\}\subseteq[a,b]\) such that \(\displaystyle \left( {\forall x \in [a,b]} \right)\left[ {f(c) \leqslant f(x) \leqslant f(d)} \right]\).
That has been called the high-point/low-point theorem.

Therefore, \(\displaystyle f(c)\left( {b - a} \right) \leqslant \int_a^b {f(x)dx} \leqslant f(d)(b - a)\) which proves it is finite.