Continuity Calculator

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

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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 xo [a, b] such that f (xo) = 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?


Checked the continuity of the given function 5x3 + 6x2 – 6 at x = 4


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

Limx → 4 f (x) = Limx → 4 5 (4)3 + Limx → 4 6 (4)2 – Limx → 4 6

Limx → 4 f (x) = 410

Limit exists.

Condition 3:

f (4) = Limx → 4 f (x)

410 = 410

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

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