How to Check if a Function Is Continuous: Precalculus Review

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Find out if a function is discontinuous

Co-authored by Jake Adams and Kyle Smith

Last Updated: September 27, 2023

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• Check for Discontinuity
|
• Checking at a Point
|
• Checking over an Interval
|
• Expert Q&A
|
• Tips

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Checking the continuity of a function is easy! The simple rule for checking is tracing your pen on the curve. If you have to pick up your pen, the function is discontinuous. We’ll review types of discontinuity and how to use limits to identify continuity at a point or over an interval. This wikiHow guide shows you how to check if a function is continuous.

Things You Should Know

• This tutorial uses a general rule (tracing) and limits to check for continuity.
• Look for point, jump, and asymptotic discontinuities in your function.
• For a point, take the limit of f(x) = f(c) for x approaches c.
• For a closed interval, you’ll need to take two limits, one for each end of the interval.

Steps

Method 1

Method 1 of 3:

Check for Discontinuity

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1. 
   1
   Look for a point discontinuity. This is also called a removable discontinuity. Discontinuities indicate that your function is not continuous. Point discontinuities occur when a point on a curve differs from the typical path of the function.

   • You’ll see a hole in the curve where the point discontinuity is. This usually looks like an unfilled circle.
   • The point will be somewhere above or below the hole.
   • For example, if you have the function f(x) = x, you expect a straight, diagonal line crossing through the origin. However, if there’s a hole in the curve at x = 3, and a point at (3, 10), the function is discontinuous.
2. 
   2
   Check for a jump discontinuity. This is when the curve suddenly jumps to another disconnected curve.

   • For example, for f(x) = x, let’s say at x = 4, the f(x) = x curve ends with a hole. Then, at x = 4, a second curve of f(x) = x + 5 begins. This second line is disconnected and above the first line.
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3. 
   3
   Identify an asymptotic discontinuity. These discontinuous functions have an asymptote that causes the parts of the function to tend toward an x value without reaching it.
4. 
   4
   Trace the curve left to right. As a general rule, a function is discontinuous if you need to pick up your pencil as you trace the curve. The next section will cover how to check continuity using limits.
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Method 2

Method 2 of 3:

Checking at a Point

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1. 
   1
   Take the limit of f(x) = f(c) for x approaches c. This limit checks both sides of a curve at a point to see if the correct f(c) value is being approached. If f(c) differs from the f(x) value you would expect at c, the function is not continuous.

   • Where:
     • f(x) is a function
     • f(c) is the function value at x = c
     • c is a given x value

Method 3

Method 3 of 3:

Checking over an Interval

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1. 
   1
   Trace the curve f(x) from a to b. For an open interval (a, b), check for discontinuity by tracing along the curve, not including a and b. If you needed to pick up your pen during the trace (e.g. for a point discontinuity), the curve is not continuous over the interval.
2. 
   2
   Check the limits for a and b. For a closed interval [a, b] you’ll need to check two limits. This is to check for holes (discontinuity) at either end of the curve.

   • Take the limit of f(x) = f(a) for x approaches a+ (a from the right).
   • Take the limit of f(x) = f(b) from x approaches b- (b from the left).
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Expert Q&A

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• Question

  How do I know when a piecewise function is continuous?

  

  Jake Adams
  Math Tutor

  Jake Adams is an academic tutor and the owner of Simplifi EDU, a Santa Monica, California based online tutoring business offering learning resources and online tutors for academic subjects K-College, SAT & ACT prep, and college admissions applications. With over 14 years of professional tutoring experience, Jake is dedicated to providing his clients the very best online tutoring experience and access to a network of excellent undergraduate and graduate-level tutors from top colleges all over the nation. Jake holds a BS in International Business and Marketing from Pepperdine University.

  

  Jake Adams

  Math Tutor

  Expert Answer

  In the context of a piecewise function, continuity is achieved when, from both the right and left approaches, the function values (f of X or Y) coincide at a specific X value. In simpler terms, the functions smoothly connect, and there is mutual agreement that a particular X value yields the same result for both functions. However, the differentiability of the piecewise function is contingent on whether the derivatives concur in terms of the values approached from both sides.

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Tips

• For general study tips, check out our guides on how to do well in Calculus and how to pass Calculus. We also have an article on understanding Calculus.
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