We have given the concept of* continuity *in this way: let F be a function defined in an interval [a, b]; if

In this way the continuity of the real functions of a real variable has been introduced only on intervals, not at points, through what is traditionally assumed as the definition of** uniform continuity**. This choice is the standard one in the field of constructive mathematics and is made in various calculus manuals (for example: Lax-Bernstein-Lax, "Calculus with Applications and Computing").

On the usual definition it will be possible to return to the following classes, focusing on the following theorem, known as the* small-span theorem*:

The concept of "uniform continuity" (in an interval) differs from that of "continuity" (in all points of the interval) only, in particular cases, on non-closed intervals.
For example, in the drawing below, F and G are uniformly continuous in each closed interval contained in the interval (a,b);
instead if we were referring to the interval (a,b) we would have that F (assuming that