Compactness and

for any `epsilon > 0`, there is a real number `delta >0` so that

for any `a` and `x` in `S`, if `|x-a| < delta` then `|f(x)-f(a)| < epsilon`.

Proposition: If `f` is uniformly continuous on and interval,`S`, then `f` is continuous on `S`.

Proof: Check the logic of the quantifiers for uniform continuity and continuity.

Example: `f(x) = 1/x` for `x \ne 0`, `f(0) = 0`; `S = (0, 1)`. `f` is continuous on `S`, but `f` is __not__ uniformly continuous on `S`.

Example: `f(x) = mx + b` is uniformly continuous for `(-oo,oo)`.

Discussion: Suppose `epsilon > 0` is given. If `m=0` then let `delta =
epsilon` will do. If `m \ne 0`, let `delta = epsilon/{|m|}`. The
verification that for any `x` and `a` if `|x-a| < delta` then
`|f(x)-f(a)| < epsilon` is left as a routine exercise for the reader.

Theorem: A continuous function on a compact set of real numbers, `K` is uniformly continuous.

Proof: Plan- Given `epsilon >0`, find an open cover of `K` and use a finite subcover of `K` to find a`delta >0`

where if `|x-a| < delta` then `|f(x)-f(a)| < epsilon`.

Start:

Consider

Using `K` is compact, there is an open subcover determined by `{a_1, a_2, ..., a_n}` with related

Then `K sub N(a_1, delta_1) uu N(a_2,delta_2) uu ... uu N(a_n,delta_n)`.

Now suppose `x,a, in K` and `|x-a| < delta`.

Then for some `j` with `1 le j le n` we have `s in N(a_j,delta_j)` or `|a-a_j|<delta_j`.

Then `|x-a_j| le |x-a| + |a-a_j| < delta + delta_j < delta_j + delta_j =2 cdot delta_{a_j}/2 = delta _{a_j} `

Thus by [1] `|f(x) - f(a_j) < epsilon/3` and `|f(a) -f(a_j)| <epsilon/3`.

So `|f(x) -f(a)| le |f(a)-f(a_j)| + |f(a_j) - f(x)| < 2 epsilon/3 < epsilon`.

Thus `f` is uniformly continuous on `K`.

EOP

For a well done proof with figures see http://pirate.shu.edu/~wachsmut/ira/cont/proofs/ctunifct.html

Theorem: Any continuous function on `[a,b]` is Darboux (or Riemann) integrable.

Proof:

To show `f` is Darboux integrable we must find a partition `P` where `U(P,f) -L(P,f) < epsilon`.

Since `f` is continuous on `[a,b]` which is compact, `f` is uniformly continous on `[a,b]`.

Choose `delta>0` so when `|x-t| < delta`, `|f(x)-f(t)|< epsilon/{b-a}`.

Then by the extreme value theorem, for any interval, `I`, with length `l(I)<delta`, `lub{f(x): x in I} - glb{f(x):x in I} = f(c_M)-f(c_m) < epsilon/{b-a}.`

Now choose N so that `Delta x = {b-a)/N <delta`, `x_k = a + k Delta x`, and `P = {x_kP: k = 0,1, ..., n}`.

Then

EOP

See also [An application of uniform continuity.]