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.