Continuity
Continuity
Definition:
 Continuity of a Function: A function $f(x)$ is continuous at a point $c$ in its domain if the following conditions hold:
 $f(c)$ is defined.
 The limit of $f(x)$ as $x$ approaches $c$ exists.
 The limit of $f(x)$ as $x$ approaches $c$ equals $f(c)$.
Types of Continuity:

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

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 $f(x)$ is continuous on an interval $[a,b]$ and $k$ lies between $f(a)$ and $f(b)$, then there exists $c$ in $(a,b)$ such that $f(c)=k$.
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.