Discontinuity of a function
Discontinuity of a Function
Definition:
 Discontinuity: A point $c$ in the domain of a function $f(x)$ where the function fails to be continuous.
Types of Discontinuities:

Removable Discontinuity:
 A discontinuity at $c$ that can be "filled in" or removed by redefining the function at that point.
 Example: $f(x)=\frac{{x}^{2}1}{x1}$, where the function is undefined but can be made continuous by defining $f(1)=2$.

Jump Discontinuity:
 A discontinuity where the function "jumps" from one value to another at a specific point.
 Example: $f(x)=\text{sign}(x)$ (signum function) at $x=0$, where the function transitions from 1 to 1 abruptly.

Infinite Discontinuity:
 A discontinuity occurring when the function's value becomes unbounded (tends to infinity) at a certain point.
 Example: $f(x)=\frac{1}{x}$ at $x=0$, where the function's values tend to infinity as $x$ approaches 0.

Essential Discontinuity:
 A discontinuity that doesn't fall into the categories of removable, jump, or infinite discontinuities.
 Example: $f(x)=\mathrm{sin}\left(\frac{1}{x}\right)$, displaying erratic oscillatory behavior around 0.
Graphical Representation:

Removable Discontinuity: The graph might have a hole or gap that can be filled by redefining the function at that point.

Jump Discontinuity: The graph displays a sudden jump from one point to another.

Infinite Discontinuity: The graph may have a vertical asymptote or an unbounded increase/decrease around a point.
Identifying Discontinuities:

Algebraic Analysis: Observing the function's behavior algebraically, like undefined values or changes in function values.

Graphical Examination: Looking for breaks, jumps, or erratic behavior in the graph of the function.