# Proof of Continuous Function

1. Dec 11, 2011

### AndersCarlos

1. The problem statement, all variables and given/known data

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].

2. Relevant 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:

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

3. 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: $$\left |f(u) - f(v) \right | \leq \left | u - v\right |$$

2. Dec 11, 2011

### 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.