A mathematical function has a discontinuity if it has a value or point that is undefined or discontinuous. Discontinuity is of utmost importance in mathematics. If you’re studying algebra, you might be required to identify the point of discontinuity in the equation.

We’ll help you figure it out by identifying the different types of discontinuity and outlining basic facts about it.

Table of Contents

## Types

### Asymptotic Discontinuity

Whenever an asymptote exists, asymptotic discontinuities occur. The asymptote of a curve in analytical geometry is a line whereby the distance between the line and the curve nears zero as both of them tend to infinity. Although at one time some sources included the requirement that the curve cannot cross the line infinitely very often, most modern authors do not subscribe to this.

If you view a graph and find it approaching a point but not exactly touching that point, this is an example of asymptotic discontinuity. Furthermore, the other side of the graph has the same thing happening to it because from both sides, you never actually see the graph touch the point. This is in essence why there is a discontinuity in the graph in the first place.

As a general rule, these asymptotes exist when the function approaches infinity at a specific value of y or x. If the function has values on an asymptote’s two sides, the connection is not possible, which means that it has to have a discontinuity at the asymptote. Always look for asymptotes at points that have a zero denominator because the value’s function is very large when the denominator gets close to zero because it becomes very small.

If you look at the fraction

### Endpoint Discontinuities

If a function is described to have a closed endpoint on its interval, it is called an endpoint discontinuity. The limit cannot exist at the endpoint, however, because the limit needs to view function values as xx approaches from both of its sizes. In other words, considering finding limx→0x−−√limx→0x (see the following graph).

In the above example, x=0x=0 is the left endpoint of the domain’s functions; i.e., [0,∞)[0,∞), and technically, the function isn’t continuous at that point because the limit doesn’t really exist. This is because xx cannot approach from the two sides.

One more thing: the function is right-hand continuous at the x=0x=0 so there are no jumps or holes at the endpoint.

### Infinite Discontinuity

Also called essential discontinuity, this occurs when you look at the domain of function and at some point, both the upper and lower limits or just one of them do not exist. It exists when one of the function’s one-sided limits is infinite; i.e., lim x → c + f ( x ) = ∞ or one of the other three varieties of the infinite limits. If the other two one-sided limits have equal values, the two-sided limit also exists.

The univariate real-valued function

The figure above is an example of the piecewise function,

a function for which both