Continuity Calculator

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

1. f is continuous on] a, b [
2. Lim _x→a⁺ f (x) = f (a)
3. 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 5x³ + 6x² – 6 at x = 4

Solution:

Condition 1:

Checking if the function is defined at x = 4

 f(x) = 5x³ + 6x² – 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.