# Continuity

## Continuity

### Definition:

• Continuity of a Function: A function $f\left(x\right)$ is continuous at a point $c$ in its domain if the following conditions hold:
1. $f\left(c\right)$ is defined.
2. The limit of $f\left(x\right)$ as $x$ approaches $c$ exists.
3. The limit of $f\left(x\right)$ as $x$ approaches $c$ equals $f\left(c\right)$.

### Types of Continuity:

• Point Continuity: A function is continuous at a specific point $c$ if ${\mathrm{lim}}_{x\to c}f\left(x\right)=f\left(c\right)$.

• Interval Continuity: A function is continuous on an interval if it's continuous at every point within that interval.

### Types of Discontinuities:

• Removable Discontinuity: A discontinuity that can be "filled in" or removed by redefining the function at that point.

• Jump Discontinuity: When the function changes abruptly from one value to another at a specific point.

• Infinite Discontinuity: Occurs when the function's value becomes unbounded (tends to infinity) at a certain point.

• Essential Discontinuity: A discontinuity that can't be classified as removable, jump, or infinite; often involves oscillatory behavior.

### Properties of Continuous Functions:

• Algebra of Continuous Functions: The sum, difference, product, and quotient of continuous functions are continuous.

• Intermediate Value Theorem: If $f\left(x\right)$ is continuous on an interval $\left[a,b\right]$ and $k$ lies between $f\left(a\right)$ and $f\left(b\right)$, then there exists $c$ in $\left(a,b\right)$ such that $f\left(c\right)=k$.

### Testing Continuity:

• Direct Substitution: Checking if the function is defined at a given point.

• Using Limits: Evaluating limits as $x$ approaches a value to verify if the function's values approach a specific value.

• Graphical Analysis: Observing the graph for any breaks, jumps, or asymptotic behavior.