8. Increasing and Decreasing Functions.—A function ƒ(x) of one variable x, defined in the interval between a and b, is “increasing throughout the interval” if, whenever x and x′ are two numbers in the interval and x′ > x, then ƒ(x′) > ƒ(x); the function “never decreases throughout the interval” if, x′ and x being as before, ƒ(x′) > ƒ(x). Similarly for decreasing functions, and for functions which never increase throughout an interval. A function which either never increases or never diminishes throughout an interval is said to be “monotonous throughout” the interval. If we take in the above definition b > a, the definition may apply to a function under the restriction that x′ is not b and x is not a; such a function is “monotonous within” the interval. In this case we have the theorem that the function (if it never decreases) has a limit on the left at b and a limit on the right at a, and these are the superior and inferior limits of its values at all points within the interval (the ends excluded); the like holds mutatis mutandis if the function never increases. If the function is monotonous throughout the interval, ƒ(b) is the greatest (or least) value of ƒ(x) in the interval; and if ƒ(b) is the limit of ƒ(x) on the left at b, such a greatest (or least) value is an example of a superior (or inferior) limit which is attained. In these cases the function tends continually to its limit.
These theorems and definitions can be extended, with obvious modifications, to the cases of a domain which is not an interval, or extends to infinite values. By means of them we arrive at sufficient, but not necessary, criteria for the existence of a limit; and these are frequently easier to apply than the general principle of convergence to a limit (§ 6), of which principle they are particular cases. For example, the function represented by x log (1/x) continually diminishes when 1/e > x > 0 and x diminishes towards zero, and it never becomes negative. It therefore has a limit on the right at x = 0. This limit is zero. The function represented by x sin (1/x) does not continually diminish towards zero as x diminishes towards zero, but is sometimes greater than zero and sometimes less than zero in any neighbourhood of x = 0, however small. Nevertheless, the function has the limit zero at x = 0.
9. Continuity of Functions.—A function ƒ(x) of one variable x is said to be continuous at a point a if (1) ƒ(x) is defined in an interval containing a; (2) ƒ(x) has a limit at a; (3) ƒ(a) is equal to this limit. The limit in question must be a limit for continuous variation, not for a restricted domain. If ƒ(x) has a limit on the left at a and ƒ(a) is equal to this limit, the function may be said to be “continuous to the left” at a; similarly the function may be “continuous to the right” at a.
A function is said to be “continuous throughout an interval” when it is continuous at every point of the interval. This implies continuity to the right at the smaller end-value and continuity to the left at the greater end-value. When these conditions at the ends are not satisfied the function is said to be continuous “within” the interval. By a “continuous function” of one variable we always mean a function which is continuous throughout an interval.
The principal properties of a continuous function are:
1. The function is practically constant throughout sufficiently small intervals. This means that, after any point a of the interval has been chosen, and any positive number ε, however small, has been specified, it is possible to find a number h, so that the difference between any two values of the function in the interval between a − h and a + h is less than ε. There is an obvious modification if a is an end-point of the interval.
2. The continuity of the function is “uniform.” This means that the number h which corresponds to any ε as in (1) may be the same at all points of the interval, or, in other words, that the numbers h which correspond to ε for different values of a have a positive inferior limit.
3. The function has a greatest value and a least value in the interval, and these are superior and inferior limits which are attained.
4. There is at least one point of the interval at which the function takes any value between its greatest and least values in the interval.
5. If the interval is unlimited towards the right (or towards the left), the function has a limit at ∞ (or at −∞).