Continuity

In simple language the word continuity refers to the property of being continuous. For example, if you have been attending your classes continuously for a month. It means you have maintained continuity in attending classes.

In mathematics, especially in calculus, the word continuity is attributed to relations. That is, if for every input of the variable, there is a defined out put, then the relation has continuity.

The graphs of these relations give a clear picture of their continuity. We say a relation has continuity if you do not see any break in its graph.

Let us study the various aspects of continuity in detail.

Continuity Definition

In the previous section we had framed the requirements for the continuity of a function at a point. Let f(x) be a function and let x = c be the point where the continuity is to be studied.

We shortlisted the conditions for the continuity at a point x = c,

1) The function should be defined at x = c.

2) The limit of the function f(x) at x = c must be defined.

But still there are lacunas left for a complete definition of continuity at a point. Now consider a function  f(x) =$\left ( \frac{x^2-1}{x-1} \right )$ and let us study the continuity at a point x = 1. For the purpose of evaluating limits, the function can be simplified as f(x) = (x + 1) and surely the limit exists at x = 1 and it is 2. But at x = 1, the function itself is not defined. Hence, under this condition the function f(x) does not have continuity at  x = 1.

Suppose the function, is redefined as, f(x) = $\left ( \frac{x^2-1}{x-1} \right )$ for x $neq$ 1, and  f(x) = x + 2  for x = 1.

Now what happens? The limit at x = 1 exists and f(1) also exists. But, f(1) = 1 + 2 = 3. Therefore, at x = 1,
the function takes a jump from 2 to 3, which is a break in continuity.

Let the function be redefined as, f(x) =$\left ( \frac{x^2-1}{x-1} \right )$ for x $\neq$ 1, and  f(x) = x + 1  for x = 1.

In this case, the limit exists at x = 1, the function is defined at x = 1, and also f(1) = limit of the function at x = 1. Therefore, now there is absolute continuity at x = 1. Thus, we are now clear with the continuity at a point and we are able to arrive at the formal definition of continuity at a point. In nutshell, the definition is as follows.

For f(x) to be continuous at x = c,

$\lim_{x\rightarrow c+}f(x)$ = $\lim_{x\rightarrow c-}f(x)$ = $f(c)$

Continuity Equation

We are aware that all mathematical relations are not functions. In the subject of mathematics, continuity equations refer to relations which are not functions but maintain continuity in their domains.
For example the equation of a circle x2 + y2 = a2 has continuity in the domain [-a, a]. Hence it is a continuity equation. Similarly, the equation of a horizontal parabola y2 = 4ax is a continuity equation because the equation is continuous in the domain, [0, infinity).

Continuity Function

A function is a one-to-one relation. Most of the applications in real life are functions. Hence is important to study the continuity of a function. In a function continuity exists, or a function is continuous if the function is defined at all points. Further, at any point, for an infinitesimal increase in the value of the variable, there must be some corresponding change in the value of the function. All polynomial functions, exponential functions, sinusoidal and cosine functions are continuous functions.  In any of these functions, the function is defined at any point.

But there are many functions in which, the function may not be defined at some points. For example, the function f(x) = $\frac{1}{x}$, is not defined when x = 0. Such functions are said to have discontinuity at such values of the variable where it they are not defined. In other words, we can term that as a break of continuity at such points.

The break of continuity in a function may be of different types. Taking the example of the same function f(x) = $\frac{1}{x}$, the function does not have the continuity only at a point x = 0. Otherwise the function is continuous for all values of x. This type of break in continuity is called as the point discontinuity. It is also called as a hole in the function at that point. This is called as removable discontinuity as the continuity can be restored by redefining the function at that point.

In some cases, the functions are defined differently in different intervals. The value of the function may not be same at the points where the interval changes.

For example if a function is defined as below,

f(x) = 2x +1 for x < 0 and f(x)  = 2x + 4 for x ≥ 0, the continuity of the function is broken at x = 0.

Also some functions at the point of discontinuity may be approaching infinity on opposite sides at particular points. For example f(x) = $\frac{1}{x}$ approaches to -$\infty$ when the variable approaches 0 from the negative side and approaches  to $\infty$ when the variable approaches from positive sides. Such a break in continuity is called  as jump discontinuity.

Continuity at a Point

While studying a function, our attention may be focused only to a small interval or a point. The first and foremost requirement is that the function should exist at that point. A graph of the function will instantly tell you about the continuity at a point. But we may not have the graphs all the times with us. Hence to study the continuity at a point, we need to apply the concept of limits at a given point.

Suppose we need to study the continuity at point x = c for a function f(x). The basic requirement of continuity at a point stipulates that f(c + h) = f(c – h) where, h is an infinitesimally small change.  In other words, the right hand side limit and the left hand side limit of the function at x = c must be same or in other words the limit of the function at x = c must exist.

In the next section, let us work out mathematically and arrive at the formal definition of continuity at a point.

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