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Not Continuous Function on Interval Calculator

Reviewed by Calculator Editorial Team

This calculator helps you determine where a function is not continuous on a given interval. Understanding discontinuities is essential in calculus and real analysis, as they reveal important properties of functions and their behavior.

What is Continuity?

A function is continuous at a point if there are no jumps, breaks, or holes at that point. Formally, a function f(x) is continuous at a point c in its domain 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 of f(x) as x approaches c equals f(c).

If any of these conditions fail, the function is discontinuous at that point. The interval of interest is where we examine the function's behavior between two points, a and b.

Types of Discontinuities

There are three main types of discontinuities:

  1. Removable discontinuity: Occurs when there's a hole in the graph. The limit exists, but the function is not defined at that point.
  2. Jump discontinuity: Occurs when the left-hand and right-hand limits exist but are not equal.
  3. Infinite discontinuity: Occurs when the function grows without bound as x approaches a certain point.

Note: A function can have multiple types of discontinuities within a single interval.

How to Find Discontinuities

To find where a function is not continuous on an interval [a, b], follow these steps:

  1. Identify all points in the interval where the function is not defined.
  2. Check for points where the limit does not exist.
  3. Look for points where the limit exists but does not equal the function value.
  4. Check for points where the function grows without bound.

The calculator automates this process by evaluating the function at many points within the interval and identifying where these conditions are violated.

Example Calculation

Consider the function f(x) = (x² - 1)/(x - 1) on the interval [-2, 2].

The function is not continuous at x = 1 because:

  • The function is undefined at x = 1 (denominator becomes zero).
  • The limit as x approaches 1 exists (it's 2).
  • The limit does not equal the function value at x = 1.

This is a removable discontinuity. The calculator would identify x = 1 as a point where the function is not continuous.

FAQ

What is the difference between a removable and non-removable discontinuity?
A removable discontinuity can be fixed by redefining the function at that point, while a non-removable discontinuity (like a jump or infinite discontinuity) cannot be fixed without changing the function's definition.
How does the calculator determine where a function is not continuous?
The calculator evaluates the function at many points within the interval and checks for points where the function is undefined, the limit doesn't exist, or the limit doesn't equal the function value.
Can a function have multiple types of discontinuities in one interval?
Yes, a function can have removable, jump, and infinite discontinuities all within the same interval. The calculator will identify all points where any type of discontinuity occurs.