Functions come in all shapes and sizes. One of the key elements of function is either the function is continuous or not? One of the biggest examples of a non-continuous function is the piece-wise function. The graph of the function breaks at a certain limit. It means that after some inputs, the function behaves differently. You can't predict anything in a piece-wise function because it breaks at some point. On the other hand, if the function is continuous, it means that the graph will have no breaking point. The function won't behave differently on any input. The indication of a continuous function is that it will go to infinity, it will be like a graph drawn without lifting the pencil from the paper. Suppose you have a function, f(x) = \sin { x }. Below is the graph of the function.

As you can see, the function keeps following a similar pattern over and over without any gap or breaking point. We will say it is a continuous function but what if someone asks you to give proof of its continuity? How to show that? Keep on reading to find it.

Continuous Function at a Point

To prove the continuity of a function, pick a point and from that point, we can prove the function's continuity. Imagine that the point you picked is x = a, now a function, f(x), is continuous at a point a, if and only if it meets the following conditions:

1. The point x = a has an image. If the image of the point "a" doesn't exist, that means the point "a" doesn't belong to a continuous graph.

\exists f(a)

2. There is a limit of the function f(x) at x = a. The limit approaches the point "a" from both sides, from the positive side to point "a" and from the negative side to point "a". If both sides are equal to \lim { x \rightarrow a } f(x) that means there is no breaking in the graph.

\exists \lim_{ x \rightarrow a } f(a) \Leftrightarrow \lim_{ x \rightarrow { a }^{ - } } f(x) = \lim_{ x \rightarrow { a }^{ + } } f(x)

3. The value of the function at the point coincides with the limit of the function at the point.

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

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Example

Study the continuity of f(x) = \left\{\begin{matrix} { x }^{ 2 } \quad if \quad x < 2 \\ 4 \quad if \quad x \geq 2 \end{matrix}\right at x = 2.

f(2) = { 2 }^{ 2 } = 4

\lim_{ x \rightarrow { 2 }^{ - } } { x }^{ 2 } = 4

\lim_{ x \rightarrow { 2 }^{ + } } 4 = 4

f(2) = \lim_{ x \rightarrow { 2 }^{ - } } { x }^{ 2 } = \lim_{ x \rightarrow { 2 }^{ + } } 4 = \lim_{ x \rightarrow 2 } f(x)

Directional Continuity

One of the conditions stated that the limit of the function approaching the point "a" should be equal to f(a), \lim_ { x \rightarrow a } f(x) = f(a). This means that the limit will be approached from 2 sides and if both are equal, it means the function is continuous. Those sides are "Left-continuous function" and "Right-continuous function".

Left-Continuous Function

The left-continuous function says that the limit of the function will be approached starting from left and ending to the point "a". It will go left to right and will finish at point "a".

f(a) = \lim_{ x \rightarrow { a }^{ - } } f(x)

f(2) = \lim_{ x \rightarrow { 2 }^{ - } } { x }^{ 2 } = 4

Right-Continuous Function

This time, the limit of the function will be approached from right to the point "a". This is called the right-continuous function.

f(a) = \lim_{ x \rightarrow { a }^{ + } } f(x)

f(2) = \lim_{ x \rightarrow { 2 }^{ + } } f(x) = 4

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Hamza

Hi! I am Hamza and I am from Pakistan. My hobbies are reading, writing and playing chess. Currently, I am a student enrolled in the Chemical Engineering Bachelor program.