structural:summarization
One-dimensional case

In one dimension, the result is intuitive and easy to prove. The continuous function f is defined on a closed interval and takes values in the same interval. Saying that this function has a fixed point amounts to saying that its graph (dark green in the figure on the right) intersects that of the function defined on the same interval which maps x to x (light green).

Intuitively, any continuous line from the left edge of the square to the right edge must necessarily intersect the green diagonal. To prove this, consider the function g which maps x to f (x) − x. It is ≥ 0 on a and ≤ 0 on b. By the intermediate value theorem, g has a zero in; this zero is a fixed point.

Brouwer is said to have expressed this as follows: "Instead of examining a surface, we will prove the theorem about a piece of string. Let us begin with the string in an unfolded state, then refold it. Let us flatten the refolded string. Again a point of the string has not changed its position with respect to its original position on the unfolded string."