Intervals Where Function Is Continuous Calculator

What Is Continuity in Functions?

A function is continuous at a point if there are no jumps, breaks, or holes at that point. More formally, a function f(x) is continuous at a point c if three conditions are met:

1. The function is defined at c (f(c) exists)
2. The limit of f(x) as x approaches c exists
3. The limit equals the function value (lim x→c f(x) = f(c))

If a function is continuous at every point in an interval, it is continuous on that interval. Points where the function is not continuous are called points of discontinuity.

How to Find Continuity Intervals

To find where a function is continuous, follow these steps:

1. Identify the domain of the function (all x-values where the function is defined)
2. Look for points where the function might be discontinuous:
   • Points where the denominator is zero (for rational functions)
   • Points where the function is not defined (like square roots of negative numbers)
   • Points where the function has a jump discontinuity
3. Check the limit of the function as x approaches each potential point of discontinuity
4. Determine if the limit equals the function value at that point
5. Combine the intervals where the function is continuous

The continuity intervals are the intervals where the function meets all three continuity conditions.

Using the Calculator

Our calculator helps you determine the continuity intervals for a given function. Simply enter the function in the provided field, and the calculator will analyze it to find where the function is continuous.

The calculator uses the following approach:

1. Parses the function to identify potential points of discontinuity
2. Checks the limit at each potential point
3. Determines if the function is continuous at each point
4. Combines the intervals where the function is continuous

Examples of Continuity Intervals

Let's look at a few examples to understand how continuity intervals work.

Example 1: Polynomial Function

Consider the function f(x) = x² + 3x + 2.

This is a polynomial function, and polynomials are continuous everywhere. Therefore, the function is continuous on the interval (-∞, ∞).

Example 2: Rational Function

Consider the function f(x) = (x² - 4)/(x - 2).

This function has a point of discontinuity at x = 2 because the denominator becomes zero. The function is continuous everywhere else, so the continuity intervals are (-∞, 2) and (2, ∞).

Example 3: Piecewise Function

Consider the piecewise function:

f(x) = {
    x + 1, if x < 0
    x², if x ≥ 0
}

This function has a point of discontinuity at x = 0 because the left-hand limit (1) does not equal the right-hand limit (0). The function is continuous on (-∞, 0) and [0, ∞).