The basic concept of continuity is a crucial part of calculus. An easy way to test it is by examining whether the pen can trace the function graph without being lifted from the paper. This is a practical way to define continuity. When we go to higher levels of concepts of continuity, a more technical approach is mandatory. 

Graphs may be continuous or broken in a few places, making them continuous or discontinuous. Function continuity indicates function properties and their function values. A function is considered continuous when the graph has no gaps or breaks at a particular interval or range, that is, with all points within that range.

Differentiability and continuity are among the most important topics and help us understand different concepts, such as continuity at specific points in time and derivation of functions.

The Basic Concept of Continuity–Examples 

• First, if the limit of f(x) when x approaches the point “a” is equal to the value of f(x) at “a”, then the function f with the variable x is at the point “a” on the real line. It’s continuous f(a).
• Second, functions are continuous if they are continuous at all points within their definition.

In calculus, the function at x = a is continuous if the following are satisfied:

• The function is defined with x = a. That is, f(a) is equal to a real number. 
• As x approaches, the function is limited. 

 Existence of  xaf(x) 

• The limit of a function when x approaches a is equal to the function value at x = a.

xa+f(x) =  xa–f(x) =f(a)

If the function is continuous at any point in the specified interval, and if f (x) is continuous in the unlimited interval (a, b), then the function is continuous in the open interval (a, b).

Interpretation of Continuity in Geometry 

A function is said to be continuous if there are no breaks in the function graph within the interval and throughout the interval.

This means that there is no break in the graph of function (c,f,(c)) function f is continuous at x = c.

Continuity in an Interval

Open Interval 

In an open interval (a.b), f(x) is continuous if, at any point in the given interval, the function is continuous.

Closed Interval

The function is continuous over a closed interval when a pencil is used to plot graph functions between two points without lifting the pencil from the paper. 

Functions are continuous from the right if xa+f(x) = f(a)

Functions are continuous from the left if xa– f(x) = f(a)

What is Discontinuity? 

Discontinuity is a condition where there is a break in the continuity of the graph. It can be classified into three: 

• Removable discontinuity- here, the discontinuity is such that there is a hole in the graph.
• Jump discontinuity- here, the different sections of the functions do not meet.
• Infinite discontinuity- here, the discontinuity is located in a vertical region. 

When f(x) is discontinuous, then 

• In a removable discontinuity – xa f(x) exists.

Here lim f(x) = L,

The real number is L.

• In a jump discontinuity xa– f(x) and xa+ f(x), both exist.

The values of +-∞ are not taken.

• In an infinite discontinuity

xa– f(x) = ±∞

Or xa+ f(x) =±∞

The Theorem of Intermediate Value

Various theorems concern functions. One such function is the theorem of intermediate value which states that over a closed bounded interval (a,b), when f is continuous, and z is a real number between f(a) and f(b), then a number c in (a,b) always satisfies f(c) = z. 

This theorem helps us to find values between f(0) and f(2). Other values cannot be found.

An example is:

Considering a function 

f(x) = (x-1)2

f(0) = 1>0

f(2) = 1>0

f(1) = 0.

Conclusion 

The importance of the basic concept of continuity lies not only in math but also finds its application in everyday life. Examples of the basic concept of continuity include: 

• In chemical laboratories, where chemicals are mixed over time to form a new compound, the formation of a new product shows a limit of a function as time reaches up to infinity.
• The various derivatives obtained help find the rate of growth of tumours. A study of the continuous and discontinuous graphs shows how fast the tumours are regressing or progressing. 
• The importance of the basic concept of continuity also lies in fixing pipes, power plants, and semiconductor technology.