# Discontinuity of a function

## Discontinuity of a Function

### Definition:

• Discontinuity: A point $c$ in the domain of a function $f\left(x\right)$ where the function fails to be continuous.

### Types of Discontinuities:

1. Removable Discontinuity:

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

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

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

• A discontinuity that doesn't fall into the categories of removable, jump, or infinite discontinuities.
• Example: $f\left(x\right)=\mathrm{sin}\left(\frac{1}{x}\right)$, 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.