Points of Discontinuity of a Function

Points of discontinuity are an important concept in the theory of continuity. Understanding their classification and properties is essential for mastering the continuity of functions.

Definition of a Point of Discontinuity

Determining a Point of Discontinuity

The function $f(x)$ is discontinuous at $x_0$ if and only if one of the following conditions holds:

1. $f(x_0)$ does not exist (the function is undefined at that point)
2. $\lim_{x \to x_0} f(x)$ does not exist
3. $\lim_{x \to x_0} f(x) \neq f(x_0)$ (the limit exists but is not equal to the function value)