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# Continuity Calculator

To use the continuity calculator, input the function, choose the variable, enter the point, and click calculate.

Table of Contents:

## Continuity calculator

Continuity calculator is find the continuity of a function at a specific point and gives you the result within seconds with steps.

## What is continuity?

A function’s continuity in mathematics indicates that there are no abrupt leaps or breaks in the graph and that the graph can be drawn without having to lift a pen. Formally speaking a function f(x)at a point x = a is said to be continuous if and only if it meets the following three criteria

*f*is continuous on] a, b [- Lim
_{x}_{→a}^{+}*f*(x) =*f*(a) - Lim
_{x}_{→b}^{- }*f*(x) =*f*(b)

## Properties of continuity:

The property of continuity has many interesting properties that are useful in analyzing functions. We state below without proof some important properties of continuous functions.

- There is at least one point x
_{o}[a, b] such that f (x_{o}) = c. This is known as the "**intermediate value theorem.**" Let f be continuous on [a, b] and c R such that f (a) c and f (b) > **Theorem of extreme values:**

According to this theorem, if f(x) is a continuous function on the range [a, b], it has a maximum and a minimum value on that range.

**Algebraic operations:**

If f (x) and g (x) are two continuous functions, then these functions are also continuous at x = a. Similarly, f (x) + g (x), f (x) - g (x), and f (x) / g (x), given g (a) 0, are likewise continuous at x = a.

**Composition:**

If both the function f (x) and the function g (x) are continuous at x = a, then their composition is also continuous.

## How to calculate continuity?

**Example**

Checked the continuity of the given function 5x^{3} + 6x^{2} – 6 at x = 4

**Solution:**

**Condition 1:**

Checking if the function is defined at x = 4

f(x) = 5x^{3} + 6x^{2} – 6

f(4) = 5 (4)^{3} + 6 (4)^{2} – 6

f(4) = 410

**Condition 2:**

Applying the limit at x → 4

Lim_{x }_{→ 4} f (x) = Lim_{x }_{→ 4 }5 (4)^{3} + Lim_{x }_{→ 4 }6 (4)^{2} – Lim_{x }_{→ 4 }6

Lim_{x }_{→ 4} f (x) = 410

Limit exists.

**Condition 3:**

f (4) = Lim_{x }_{→ 4} f (x)

410 = 410

So, this function satisfied all conditions of continuity thus this function is continuous. ** **