How do you find the points of continuity and the points of discontinuity for a function?

Question

Paisley Johnson · Accepted Answer

A simple statement can be made as follows:

The points of continuity are points where a function exists, that it has some real value at that point. Since the question emanates from the topic of 'Limits' it can be further added that a function exist at a point 'a' if #lim_ (x->a) f(x)# exists (means it has some real value.) The points of discontinuity are that where a function does not exist or it is undefined.

Ezra Adams · Answer

undefined To find the points of continuity for a function, we need to check three conditions: 
1. The function must be defined at that point.
2. The limit of the function as x approaches that point must exist.
3. The value of the function at that point must be equal to the limit.

To find the points of discontinuity for a function, we need to identify any points where one or more of the above conditions are not met. Discontinuities can be classified into three types: 
1. Removable discontinuity: The limit exists, but the value of the function at that point is different from the limit.
2. Jump discontinuity: The limit exists, but the value of the function "jumps" from one side of the point to the other.
3. Essential discontinuity: The limit does not exist at that point.

By analyzing the function and applying these conditions, we can determine the points of continuity and discontinuity.