For , we have the following theorem.

Banach’s Contractive Mapping Theorem:

Every contractive mapping (i.e., there is such that for all , in ) has a unique fixed point , i.e., .

For if is continuous, and attains both positive and negative values (i.e., and for some , in ), then has a fixed point. This follows from the Intermediate-value Theorem.

Therefore do not worry about tomorrow, for tomorrow will worry about itself. Each day has enough trouble of its own.