What is the proof that a contraction mapping is a continuous function?

Answer 1

First, we'll start with the definition of a contraction mapping:

Given the metric space (B, d), with x and y Є B, and metric d(x, y) = |y - x|; a function f : B → B is a contraction if there exists some real number k < 1 such that,

d[f(x), f(y)] ≤ kd(x, y).

All that the above is saying is that |f(y) - f(x)| ≤ k|y - x|, or that the distance between y and x that's been scaled down by a factor of k, is greater than or equal to the distance between y and x after the function f has been acted on them. Even simpler; the distance between y and x is less after the function's acted on them than before it had. Hence, the term contraction.

Now, we'll introduce two new variables. Let δ = ε with ε > 0.

If d(x, y) < δ, then d[f(x), f(y)] ≤ kδ. Since k < 1 that means that d[f(x), f(y)] < ε.

Well, that's the definition of a continuous function at the point y, but since y is an arbitrary value, f is continuous on B.

Q.E.D.