Continuity of function

The continuity of a function, as intuition tells us, tells us whether the function is continuous at a given point or if it is somehow disjoint. We define and calculate continuity using limit functions, if you don't know them, they won't catch on.

Definition of the continuity of a function

Continuity is surprisingly quite simple to define and easy to understand if you fully understand limits. So the definition of function continuity:

A function f(x) is called continuous at a point a of the definition domain of a function f, if:

$$ \lim_{x\rightarrow a} f(x) = f(a) $$

We can still define left and right continuity if we substitute left or right after limit. Notice:

• We have so far only defined the definition of the continuity of the function f at the point. If the function f is continuous over all points of its definitional domain, then we can say that the function f is continuous over the entire definitional domain.
• The point a in the definition of continuity must be a mass point.
• For a function to be continuous at the point a, f(a) must exist (the value of a must be from the definitional domain of the function), the limit $\lim_{x\rightarrow a} f(x)$ must exist, and both must be equal to the same value.
• A function continuous at a point is continuous from left and right at that point. A function, as was the case with limits, can only be continuous at a point from the left or from the right. If the limit is different on the left and right, the funke is not continuous at that point.

What does the definition tell us? First, it must be true that the function is defined at the point. If a funke is not defined at a given point, it is certainly not continuous at that point. If it is defined, we look at the limits. If the function approaches the function value at a given point from the right and from the left, then the function is continuous at that point. If the limits are different from the functional value, then the function is discontinuous at that point.

A simple example: in the function limit article we were shown the function $f(x)=\frac{x}{3}$, it is a simple linear function. For any point a∈ ℝ, it is true that

$$ \lim_{x\rightarrow a} f(x) = f(a). $$

Thus the function f is continuous over its entire definitional domain. For example, if we put the number 6 after a, then we would have:

\begin{eqnarray} f(6) &=& 2\lim_{x\rightarrow 6} \frac{x}{3} &=& 2\end{eqnarray}

Discontinuity points

If a function is not continuous at a point a, it means that it is discontinuous at that point. This is not a big surprise. However, there are several kinds of discontinuity points, depending on how much we would have to "modify the graph of the function" to make the function continuous.

We have three basic kinds of discontinuity:

A point of removable discontinuity

We call a point of removable discontinuity a point for which both one-sided limits exist, these limits are equal, i.e. the function has a limit at that point but this limit is different from the function value or the function is not defined at that point. If a is the point of removable discontinuity of the function f(x), then:

$$ \lim_{x\rightarrow a^+} f(x) = \lim_{x\rightarrow a^-} f(x), \quad \mbox{ But }\quad \lim_{x\rightarrow a^+} f(x) \ne f(a) \ne \lim_{x\rightarrow a^-} f(x) $$

As an example, we show the absolute value function from the signum x. We want to know whether the function is continuous at the point a = 0. The graph follows:

We can see that the function is almost entirely continuous, only at the point zero is there a gap, because the function value here is zero, not one. Yet the limits on the left and right are equal to one at this point. This discontinuity is called removable, because by simply redefining that single point we get a continuous function. This discontinuity point is thus easily removable.

A discontinuity point of the first kind

The point of discontinuity of the first kind is called the point a, at which both one-sided limits exist, these are also proper (finite) but not equal. Definition:

$$ \lim_{x\rightarrow a^+} f(x) \ne \lim_{x\rightarrow a^-} f(x) $$

We will illustrate this discontinuity point with a simple function f(x) = sgn(x):

Again we see that at the point a = 0 the function is discontinuous. In doing so, the limit on the left is equal to −1 and the limit on the right is equal to 1:

\begin{eqnarray} \lim_{x\rightarrow0^-} \mbox{sgn}(x) &=& -1 \lim_{x\rightarrow0^+} \mbox{sgn}(x) &=& 1 \end{eqnarray}

The limits on the left and right are not equal, but are proper (finite). Thus, this is a discontinuity point of the first kind.

For this discontinuity point, we also define the notion of a function jump at the point. It is the difference of these limits, so the jump s of the function f at the point a is the value:

$$ s = \lim_{x\rightarrow0^+} f(x) - \lim_{x\rightarrow0^-} f(x) $$

A discontinuity point of the second kind

We call a point of discontinuity of the second kind a point that has at least one non-proper (infinite) one-sided limit, or if at least one limit does not exist. For an example, we show the function $f(x)=\frac{1}{x}$. The discontinuity point is at point zero:

The limit on the left at point zero is minus infinity and the limit on the right at point zero is plus infinity. Both limits are non-proprietary, so this is a discontinuity point of the second kind. This discontinuity cannot be removed in any simple way.

Continuity on the interval

We already have a defined continuity at a point, but we do not have a defined continuity on an interval. So: a function is continuous on an interval I, if it is continuous at every point on that interval. It is important to carefully distinguish between an open and a closed interval. If we have an open interval on both sides, the function need not be continuous at the extreme points - just by the principle that an open interval need not have any extreme point. However, if we have a closed interval, the function must be continuous at these extreme points on the left or right.

We haven't had a signum function anywhere for a long time. So let's have a signum function. On an open interval (0, ∞), the function is continuous. The signum function does have a discontinuity point at zero, but the openness of the interval makes zero exempt from testing. On the interval <0, ∞), the function is no longer continuous because it is not right-continuous at point 0.

And why is it not right-continuous? Because the limit from the right is equal to one, but the functional value is equal to zero. The values are not equal, so the function is not continuous at that point from the right.

We call a function piecewise continuous if it contains a finite number of discontinuity points of the first kind (or a removable discontinuity). Thus, if it contains infinitely many discontinuity points, it is not piecewise continuous; beware of this.

In practice, we find the continuity of a function by using the derivative of the function. A function that is derivable at a point is also continuous at that point.