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.
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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 , 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, has a discontinuity at (where the denominator vanishes), but a look at the plot shows that it can be filled with a value of . Put formally, a real-valued univariate function is said to have a removable discontinuity at a point in its domain provided that both and 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 has jump discontinuities at the integers; at , it jumps from (the limit approaching from the left) to (the limit approaching from the right). A real-valued univariate function has a jump discontinuity at a point in its domain provided that and both exist, are finite and that .
A third type is an infinite discontinuity. A real-valued univariate function is said to have an infinite discontinuity at a point in its domain provided that either (or both) of the lower or upper limits of goes to positive or negative infinity as tends to . For example, (from our "removable discontinuity" example) has an infinite discontinuity at . To the right of , 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.