Continuous Function and Integral Proof

[AndersCarlos]

Homework Statement

Let f be a function such that:

\left |f(u) - f(v) \right | \leq \left | u - v\right |

for all 'u' and 'v' in an interval [a, b].

a) Prove that f is continuous at each point of [a, b]
b) Assume that f is integrable on [a, b]. Prove that:
\left | \int_{a}^{b} f(x)dx - (b - a)f(c)\right | \leq \frac{(b-a)^{2}}{2}

for any 'c' in [a, b].

Homework Equations

a)

I considered that:

\left | f(u) - f(v) \right | < \epsilon
\left | u - v \right | < \delta

b)

I rewrote (b - a)f(c) as:

\int_{a}^{b} f(c)dx

The Attempt at a Solution

a) I've attempted to consider \delta \geq \epsilon Since the absolute value of the difference between 'u' and 'v' will be greater or equal to the absolute value of the difference between the difference of 'f(u)' and 'f(v)'. But no conclusive proof without supposing was achieved.

b) Maybe considering g(x) = \int_{a}^{b} f(x)dx So as the integral can be considered as a function, I could use relation: \left |f(u) - f(v) \right | \leq \left | u - v\right |

 

[AndersCarlos]

Just a little information: This is question 33, section 3.6, chapter 3 from Apostol Vol. 1. I would really appreciate any help. I know that maybe no one will answer, but thanks anyway.