Interval of Continuity Calculator

Determine the interval of continuity for a function using our calculator. A function is continuous on an interval if it has no jumps, breaks, or holes within that interval. This tool helps identify where a function remains continuous based on its definition and any points of discontinuity.

What is Continuity in Mathematics?

Continuity is a fundamental concept in calculus and analysis. A function f(x) is continuous at a point x = a if three conditions are met:

The function is defined at x = a (f(a) exists)
The limit of f(x) as x approaches a exists
The limit equals the function value (lim(x→a) f(x) = f(a))

A function is continuous on an interval if it's continuous at every point within that interval. Points where continuity fails are called points of discontinuity.

Note: A function can be continuous everywhere except at isolated points. For example, f(x) = 1/x is continuous everywhere except at x = 0.

How to Find Intervals of Continuity

To determine where a function is continuous:

Identify any points where the function is undefined
Find points where the limit does not exist
Check for points where the limit exists but doesn't equal the function value
Combine these points to determine intervals where continuity holds

For a rational function f(x) = P(x)/Q(x), the interval of continuity is all real numbers except where Q(x) = 0.

Step-by-Step Process

Write down the function definition
Identify any denominators or square roots that could cause discontinuities
Find all x-values that make denominators zero or expressions under roots negative
Exclude these points from the interval
State the interval in interval notation

Example Calculation

Let's find the interval of continuity for f(x) = (x² - 4)/(x - 2).

The function is undefined when the denominator is zero: x - 2 = 0 → x = 2
The limit as x approaches 2 exists but is infinite
Therefore, the function is continuous everywhere except at x = 2

The interval of continuity is: (-∞, 2) ∪ (2, ∞)

This function has a vertical asymptote at x = 2, which is a point of discontinuity.

Common Functions and Their Continuity

Here are some common functions and their intervals of continuity:

Function	Interval of Continuity	Points of Discontinuity
f(x) = x²	(-∞, ∞)	None
f(x) = 1/x	(-∞, 0) ∪ (0, ∞)	x = 0
f(x) = √x	[0, ∞)	All x < 0
f(x) = sin(x)	(-∞, ∞)	None

FAQ

What is the difference between continuity and differentiability?

All differentiable functions are continuous, but not all continuous functions are differentiable. A function is differentiable at a point if it has a defined derivative at that point, which requires the function to be both continuous and smooth at that point.

Can a function be continuous on an open interval but not closed?

Yes, a function can be continuous on an open interval (a,b) but not at the endpoints a or b. For example, f(x) = 1/x is continuous on (0,∞) but not at x=0.

How do I find the interval of continuity for piecewise functions?

For piecewise functions, check each piece separately and identify where the function is defined and continuous. The overall interval of continuity is the union of all intervals where each piece is continuous.