Now that we've defined the limit of a function, we're in a position to define what it means for a function to be continuous. The notion of Continuity captures the intuitive picture of a function "having no sudden jumps or oscillations". We will see several examples of discontinuous functions that illustrate the meaning of the definition. The idea of continuous functions is found in several areas of mathematics, apart from real analysis.
We say that is continuous at if and only if
For every , there exists such that .
We say itself is continuous if this condition holds for all points in .
If is a union of intervals, the statement is equivalent to saying that .
Since limits are preserved under algebraic operations, we see that if and are both continuous at c:
- is continuous at for all .
- is continuous at .
- is continuous at .
- is continuous at , assuming is non-zero.
We can use sequential limits to prove that functions are discontinuous as follows:
- is discontinuous at if and only if there are two sequences and such that .
Another result that will allow us to construct many examples of continuous functions is that any composition of continuous functions is itself continuous:
If and are continuous, then the composition is continuous on A.
Let . Let .
Since f is continuous, .
Since g is continuous, .
Thus , so is continuous on A.
The Intermediate Value Theorem
This is the big theorem on continuity. Essentially it says that continuous functions have no sudden jumps or breaks.
Theorem (Intermediate Value Theorem)
Let f(x) be a continuous function. If and , then .
Let , and let .
Let . By continuity, .
If f(c) < m, then , so . But then , which implies that c is not an upper bound for S, a contradiction.
If f(c) > m, then since , . But since , , so = m, which implies that , a contradiction.
We will now prove the Minimum-Maximum theorem, which is another significant result that is related to continuity. Essentially, it states that any continuous image of a closed interval is bounded, and also that it attains these bounds.
Minimum - Maximum Theorem
Let be continuous
(i) is bounded
(ii)If are respectively the upper and lower bounds of , then there exist such that
(i)Assume if possible that is unbounded.
Let . Then, is unbounded on at least one of the closed intervals and (for otherwise, would be bounded on contradicting the assumption). Call this interval .
Similarly, partition into two closed intervals and let be the one on which is unbounded.
Thus we have a sequence of nested closed intervals such that is unbounded on each of them.
We know that the intersection of a sequence of nested closed intervals is nonempty. Hence, let
As is continuous at , there exists such that But by definition, there always exists such that , contradicting the assumption that is unbounded over . Thus, is bounded over
(ii) Assume if possible, but .
Consider the function . By algebraic properties of continuity, is continuous. However, being a cluster point of , is unbounded over , contradicting (i). Hence, . Similarly, we can show that .
As mentioned, the idea of continuous functions is used in several areas of mathematics, most notably in Topology. A different characterization of continuity is useful in such scenarios.
is continuous at if and only if for every open neighbourhood of , there exists an open neighbourhood of such that
It must be mentioned here that the term "Open Set" can be defined in much more general settings than the set of reals or even metric spaces, and hence the utility of this characterization.
We say that is Uniformly Continuous on if and only if for every there exists such that if and then
We say that is Lipschitz continuous on if and only if there exists a positive real constant such that, for all , .
The smallest such is called the Lipschitz constant of the function .