# Continuity and Continuous of a Function in Mathematics.

*I am a PhD student of mathematics. I have complete MS in math from the University of Pakistan and have been writing online since 2020.*

## Continuity in Mathematics

## Continuity and Continuous of a Function in Mathematics.

**Introduction:**

The formal description of the matter of some debate of a function that changes over time without any sharp breaks or leaps is called continuity in mathematics. The property of a function known as continuity is what makes a graphical representation of a function into a continuous wave.

The modern concept of the limit of a function goes all the way back to Bolzano, who introduced the fundamentals of the epsilon-delta technique to construct continuous functions in 1817, even if it was implicit in the development of calculus in the 17th and 18th century.

We must first establish continuity at an interior point (which includes a two-sided limit) and continuity at an endpoint in order to define continuity at a point in the domain of a function (which involves a one-sided limit).

**Definition of Continuity:**

**Continuity at Interior Point:**

A function y = f(x) is continuous at an interior point c of its domain if,

lim x→c f(x) = f(c)

**Continuity at Endpoint:**

A function z = g(x) if, it is continuous at either the right endpoint b or the left endpoint an of its domain if,

lim x→ a^{+} g(x) = g(a) or lim x→ b^{-} g(x) = g(b)

**Continuity from the Right-Side or Right Continuous:**

An area of a domain of function is said to be right-continuous (continuous from the right-hand side) if,

lim x→ a^{+} g(x) = g(a)

**Continuity from the Left-Hand Side or Left Continuous:**

An area of a domain of function is said to be left-continuous (continuous from the left-hand side) if,

lim x→ b^{-} g(x) = g(b)

So, if a function is right-continuous at its right endpoint ** a **and left-continuous at its right endpoint

*, then it is continuous at both of these endpoints, a function is considered continuous at an interior point c at its domain, If and only if it is both right-continuous and left-continuous at c.*

**b****EXAMPLES:**

**Example No.1:**

The function g(x) =√4- x^{2} t is continuous over the whole of its domain with the possible exception of [-2, 2]. provide both x = 2 and x = -2, where g(x) is left continuous, and g(x) is right continuous respectively.

**Example No.2:**

At x = 0, the unit step function U(x) is right continuous, but it is neither left continuous or doesn't otherwise satisfy the definition of continuousness, hence it is not continuous at that point. At x = 0, there is a jump discontinuity.

A continuity at a point can be summed up as a continuity test.

**Continuity Test:**

A function g(x) is continuous at an interior point x = c of its domain. If and only if it satisfies the three criteria listed below,

- g(c) exists (c is a part of region.)
- Lim x→ c g(x) exists (g has some limit as x→ c)
- Lim x→ c g(x) = g(c) (the function value is equal to the limit)

The points 2 and 3 limitations for one-sided continuity and continuity at an endpoint should be changed for the relevant one-sided limits.

**EXAMPLES:**

**Example No.1**:

T The function y = 1 / x is a continuous function. so, it is continuous over its entire domain. It is undefined at x = 0, so it causes discontinuity at x = 0, when x = 0 is applied to y. On any interval that contains x = 0, it is discontinuous.

**Example No.2:**

Everywhere, the identity function and constant functions are continuous.

**Important Note:**

Algebraic functions and its combinations are continuous.

**Continuous Functions:**

**Definition:**

If a function is continuous on an interval, it must also be continuous at every point along the interval. A function that is continuous throughout its entire domain is said to be continuous. A continuous function does not have to be continuous at all times.

**EXAMPLES:**

** Example No.1:**

The function y = 1 / x is a continuous function. Because it is continuous over its entire domain. But it is undefined at 0, it causes discontinuity at x = 0 when x = 0 is applied to y. On any interval that contains x = 0, it gives discontinuous.

** Example No.2:**

The identity functions and constant functions are continuous functions everywhere on its entire domain.

**Important Note:**

Algebraic combinations of continuous functions are continuous wherever they are defined.

**Characteristics of Continuous Functions:**

There are 7 properties of Continuous Functions:

**1**. **Sum Property of Continuous Functions:**

If the functions g and h are continuous at x = c then its sum is defined as,

g + h

*2.* **Difference property of Continuous Functions:**

If the functions g and h are continuous at x = c then its difference property is defined as,

g- h

**3.** **Constant Multiple Property of Continuous Functions:**

If the functions g and h are continuous at x = c then its constant multiple property defined as,

k. g where k is any constant.

**4. Product Property of Continues Function:**

If the functions g and h are continuous at x = c then its product property can be defined as,

g. h

**5. Quotient Property of Continuous functions:**

If the functions g and h are continuous at x = c then its Quotient property can be defined as,

g / h where h is not equal to zero.

**6. Power Property of Continuous Functions:**

If the functions g is continuous at x = c then its power property is defined as,

^{g n}

**7. ****Root Property of Continuous functions:**

If the functions g is continuous at x = c then it roots property is defined as,

^{g1}/n it is defined on an open interval containing c, where n is a positive integer.

**Example:**

The polynomials are continuous functions, if p(x) and Q(x) are two polynomials. The quotient P(x) / Q(x) is a continuous function as a result.

https://discover.hubpages.com/education/Limit-in-Mathematics

** References**:

Weir, M.D., Thomas, G.B., Hass, J. and Giordano, F.R., 2005. *Thomas' calculus*. Pearson Education India.

*This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.*

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