Continuous and Discontinuous function
Function are the relation between input to output value. As we see functions in calculus are not a small thing to explain in a single line but it may be categorized into two main types. These types are continuous and discontinuous function. Let's study these easy concepts about functions.
Continuous Functions:
A function is said to be continuous if it has a smooth and single curve line without any disconnection.
If we say in general it is that function that can be drawn without having to lift a pencil on the paper as long as the pencil is in contact with the paper we can trace a continuous curve.
No doubt, it is not a standard definition of continuous function but it is a good way to understand its concept.
A function can be continuous only for the domain of that function.
Let’s see an example for a continuous function
Continuous Function Example:
Consider we have a function y= sin(x)
In this case, the domain of that function is all Real numbers. So the graph obtained should be continuous in this way:
So, we can see a smooth graph without any breakage which is basically called a continuous function.
Discontinuous function:
In the discontinuous function, we will find gaps, or perhaps a better word would be broken.
In the graph of a discontinuous function, the graph is all tidy and connected from the left to the right until we reach at some point x1 where it breaks and restarts from someplace below it, then it rises higher above to this point before breaking again at x2 and so on as you see below.
Let's study this graph in short, these are the points on the x-axis i.e x1,x2,x3 that is the domain of the function where the function is discontinuous we call them points of discontinuity. One more interesting thing to observe is this portion of the graph consists of just one point so at x3 the graph actually breaks twice.
Moreover, we also find little filled in circles in the corners, and also there are unfilled circles above or below filled ones what do they represent?
We can elaborate it using a simple example
Discontinuous Function Example:
Consider we have a function
$$ f(x) = \begin{cases} 2x, & \text{when} \; -1 \;\le\; x \;\le\; 2 \\ x, & \text{when} \;\; 2 \;\le\; x \;\le\; 4 \end{cases}$$
We obtain a graph from a given function
There appears to be a big gap in the graph between the numbers 2 and 4 on the x-axis. In fact, there is no plot between these two points as the function is not defined for the numbers between 2 and 4.
The function is only defined for x's between negative 1 and 2 including at negative 1 and 2 and also between 4 and 6 including both 4 and 6.
These including negative 1 and 2 will create a filled hole but if we have a function like $$2x, \;\; \text{when} \;\; -1 \;\le\; x \;\le\; 2 $$.
In this condition, the function should have a filled circle at negative 1 but an unfilled circle at 2 because 2 is not included here.
Condition for not continuous:
A function may be not continuous or discontinuous in the following three conditions:
- Hole in a curve.
- Jump between curves.
- Asympotate curves.