# Continuity and Discontinuity

Functions with graphs drawn without lifting the pencil are special due to continuity. This property defines how smooth a function is. Explore continuity, types, conditions, and examples in this article.

## What is continuity and its conditions?

Function is said to continous in a given space if there is no separation of the activity graph throughout the interval. Assume that the “f” function is a set of real numbers and that “c” is a point in Domain of f. Then f continues to c if

lim x → c f (x) = f (c)

Condition For Continuity The function must be defined at the point. Its limit at the point must exist. The value of the function at that point must equal its limit at that point.

Or in other words,

A function is said to be continuous at x = c where c belongs to the domain of f(x) if

lim x → c- f (x) = lim x → c+ = f (c)

i.e., LHL = RHL = Value of function at x = c

### Condition For Continuity

The function must be defined at the point.

Its limit at the point must exist.

The value of the function at that point must equal its limit at that point.

## What are discontinuity and its types?

In mathematics, a function is termed discontinuous when its continuity breaks. A graph where points aren't connected signifies discontinuity. Specifically, a function f(x) has a first-kind discontinuity at x=a if both its left-hand and right-hand limits exist but aren't equal. From the left, it occurs if the left-hand limit exists but isn't equal to f(a).

Removable Discontinuity A removable discontinuity arises in rational expressions when numerator and denominator share common factors. Function f(x) exhibits this at x=a if the left-hand limit equals the right-hand limit at x approaches a, yet differs from f(a).

### Types of Discontinuities

There are three different types of Discontinuities:

### Removable Discontinuity

A removable discontinuity arises in rational expressions when numerator and denominator share common factors. Function f(x) exhibits this at x=a if the left-hand limit equals the right-hand limit at x approaches a, yet differs from f(a).

### Jump Discontinuity

Discontinuity of first kind: functions f (x) is said to have a discontinuity of the first kind from the right at x=a,if the right hand of the function exists but is not equal to f(a).In jump discontinuity ,the left hand limit and the right hand limit exists and are finite but not equal to each other .

Discontinuity of second kind: A function f(x) is said to have discontinuity of the second kind at x=a ,if neither left hand limit of f(x) at x=a nor right-hand limit of f(x) at x=a exists.

A jump discontinuity is when the function jumps from one location to another.

### Infinite Discontinuity

With continuity, the function varies from x = a to provide a non-continuous environment. It means that the function f (a) is not defined. Since the value of the function x = a does not come close to any value or is inclined to infinity, the limit of activity x → a is also not defined.Infinity discontinuity also known as asymptote discontinuity.

The graph below shows a vertical asymptote that makes the graph discontinuous, because the function exists on both sides of the vertical asymptote.

On the other hand, the vertical asymptote in this graph is not a point of discontinuity, because it doesn’t break up any part of the graph.

### Conclusion

In the above article discussed the meaning of continuity and discontinuity, examples, types, and key concepts for solving right-hand and left-hand limits continuity and discontinuity.

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## Frequently Asked Questions on Continuity and Discontinuity

**Ans. **Continuity refers to the smoothness of a function without breaks, while discontinuity indicates points where the function breaks or is undefined.

**Ans.** A continuous function has no breaks or jumps, while a discontinuous function has points where it is not continuous.

**Ans. **Continuity describes the smoothness of a function, whereas a continuous function is one that is smooth without breaks or jumps.

**Ans.** In calculus, discontinuity refers to points in a function's domain where it fails to be continuous, often due to jumps, holes, or asymptotes.