Online Discontinuity Calculator - Wolfram|Alpha

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More than just an online tool to explore the continuity of functions

Wolfram|Alpha is a great tool for finding discontinuities of a function. It also shows the step-by-step solution, plots of the function and the domain and range.

Learn more about:

• Discontinuities »

Tips for entering queries

Enter your queries using plain English. To avoid ambiguous queries, make sure to use parentheses where necessary. Here are some examples illustrating how to ask for discontinuities.

• discontinuities of (x+4)/x
• discontinuities of (x^2+1)/(x^2-1)

• 1/(e^(1/x)-1) discontinuities
• floor(x) discontinuous

• View more examples »

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What are discontinuities?

A discontinuity is a point at which a mathematical function is not continuous.

Given a one-variable, real-valued function y = f (x)y=fx, there are many discontinuities that can occur. The simplest type is called a removable discontinuity. Informally, the graph has a "hole" that can be "plugged." For example, f (x) = Start Fraction, Start numerator, x-1 , numerator End,Start denominator, Start Power, Start base, x , base End,Start exponent, 2 , exponent End , Power End -1 , denominator End , Fraction Endfx=x-1x2-1  has a discontinuity at x = 1x=1 (where the denominator vanishes), but a look at the plot shows that it can be filled with a value of Start Fraction, Start numerator, 1 , numerator End,Start denominator, 2 , denominator End , Fraction End12 . Put formally, a real-valued univariate function y = f (x)y=fx is said to have a removable discontinuity at a point Start subscript, Start base, x , base End,Start subscript, 0 , subscript End , subscript Endx0 in its domain provided that both f ( Start subscript, Start base, x , base End,Start subscript, 0 , subscript End , subscript End )fx0 and Start Limit, Start variable, x , variable End,Start target value, Start subscript, Start base, x , base End,Start subscript, 0 , subscript End , subscript End , target value End,Start expression, f (x) , expression End , Limit End = L < ∞limxmm-template-arrow-right-8x0fx=L<∞ exist.

Another type of discontinuity is referred to as a jump discontinuity. Informally, the function approaches different limits from either side of the discontinuity. For example, the floor function f (x) = ⌊x⌋fx=x has jump discontinuities at the integers; at x = 2x=2, it jumps from 11 (the limit approaching from the left) to 22 (the limit approaching from the right). A real-valued univariate function y = f (x)y=fx has a jump discontinuity at a point Start subscript, Start base, x , base End,Start subscript, 0 , subscript End , subscript Endx0 in its domain provided that Start Limit from the Left, Start variable, x , variable End,Start target value, Start subscript, Start base, x , base End,Start subscript, 0 , subscript End , subscript End , target value End,Start expression, f (x) , expression End , Limit from the Left End = Start subscript, Start base, L , base End,Start subscript, 1 , subscript End , subscript Endlimxmm-template-arrow-right-8x0-fx=L1 and Start Limit from the Right, Start variable, x , variable End,Start target value, Start subscript, Start base, x , base End,Start subscript, 0 , subscript End , subscript End , target value End,Start expression, f (x) , expression End , Limit from the Right End = Start subscript, Start base, L , base End,Start subscript, 2 , subscript End , subscript Endlimxmm-template-arrow-right-8x0+fx=L2 both exist, are finite and that Start Not Equal, Start left-hand side, Start subscript, Start base, L , base End,Start subscript, 1 , subscript End , subscript End , left-hand side End,Start right-hand side, Start subscript, Start base, L , base End,Start subscript, 2 , subscript End , subscript End , right-hand side End , Not Equal EndL1≠L2.

A third type is an infinite discontinuity. A real-valued univariate function y = f (x)y=fx is said to have an infinite discontinuity at a point Start subscript, Start base, x , base End,Start subscript, 0 , subscript End , subscript Endx0 in its domain provided that either (or both) of the lower or upper limits of ff goes to positive or negative infinity as xx tends to Start subscript, Start base, x , base End,Start subscript, 0 , subscript End , subscript Endx0. For example, f (x) = Start Fraction, Start numerator, x-1 , numerator End,Start denominator, Start Power, Start base, x , base End,Start exponent, 2 , exponent End , Power End -1 , denominator End , Fraction Endfx=x-1x2-1  (from our "removable discontinuity" example) has an infinite discontinuity at x = −1x=−1. To the right of -1-1, the graph goes to ∞∞, and to the left it goes to -∞-∞.

There are further features that distinguish in finer ways between various discontinuity types. They involve, for example, rate of growth of infinite discontinuities, existence of integrals that go through the point(s) of discontinuity, behavior of the function near the discontinuity if extended to complex values, existence of Fourier transforms and more.