Function is continuous at some point , if the following conditions are hold:

I.e., the limit of the function if (from left), equals to the limit of the function if (from the right) and equals to value of the function at the point .

If at least one condition is violated, then the function has a discontinuity at the point .

All
*points of discontinuities*
are divided to points of discontinuities of first and second kind.

If exist finite one-sided limits
and
, then the point
is called the discontinuity point of
*first kind*.

The discontinuities points of the first kind are in turn subdivided into the points of removable discontinuities and the jumps.

If
is discontinuity point of the first kind and the
, the point
is called the point of
*removable discontinuity*.

The plot of corresponding function is given below:

If
, then the function has a
*jump*
at the point .
The value of the jump is calculated by the formula
. The corresponding graph is shown in the figure:

If at least one of the limits
or
is equal to
, the point
is called discontinuity point of
*second kind*. An example of the corresponding function graph is shown in the figure below:

Our online calculator, built on the basis of the Wolfram Alpha system, calculates the discontinuities points of the given function with step by step solution.

Discontinuities calculator

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