Continuity and Differentability
Continuity:

Definition: A function $f(x)$ is continuous at a point $c$ if three 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 Discontinuities:
 Removable Discontinuity: A discontinuity that can be filled by redefining the function at that point.
 Jump Discontinuity: The function "jumps" from one value to another at a specific point.
 Infinite Discontinuity: When the function's value becomes infinite at a certain point.
 Essential Discontinuity: A discontinuity that cannot be classified as removable, jump, or infinite.

Continuity of Composite Functions:
 The composition of continuous functions is continuous.
 $f(g(x))$ is continuous at $x=c$ if both $f(x)$ and $g(x)$ are continuous at $x=c$.
Differentiability:

Definition: A function $f(x)$ is differentiable at a point $c$ if the derivative ${f}^{\mathrm{\prime}}(c)$ exists.
 The derivative represents the rate of change of a function at a given point.

Conditions for Differentiability:
 Existence of Derivative: The limit ${\mathrm{lim}}_{h\to 0}\frac{f(c+h)f(c)}{h}$ exists.
 Continuity at the Point: If a function is differentiable at $c$, it must also be continuous at $c$.

Differentiability Rules:
 Sum/Difference Rule: $(f\pm g{)}^{\mathrm{\prime}}(x)={f}^{\mathrm{\prime}}(x)\pm {g}^{\mathrm{\prime}}(x)$
 Product Rule: $(f\cdot g{)}^{\mathrm{\prime}}(x)={f}^{\mathrm{\prime}}(x)\cdot g(x)+f(x)\cdot {g}^{\mathrm{\prime}}(x)$
 Quotient Rule: ${\left(\frac{f}{g}\right)}^{\mathrm{\prime}}(x)=\frac{{f}^{\mathrm{\prime}}(x)\cdot g(x)f(x)\cdot {g}^{\mathrm{\prime}}(x)}{(g(x){)}^{2}}$
 Chain Rule: $(f(g(x)){)}^{\mathrm{\prime}}={f}^{\mathrm{\prime}}(g(x))\cdot {g}^{\mathrm{\prime}}(x)$

Differentiability and Continuity:
 Differentiability implies continuity, but continuity does not necessarily imply differentiability.
 A function can be continuous at a point where it is not differentiable, such as at sharp corners or cusps.
Key Concepts:
 Local Linearity: Differentiability implies that the function can be approximated by a straight line at that point.
 Critical Points: Points where the derivative is zero or undefined; they can indicate maxima, minima, or points of inflection.
Testing Continuity and Differentiability:
 Using Limits: Calculate limits to check continuity and differentiability at a point.
 Graphical Analysis: Observing the graph to identify jumps, holes, or sharp turns to determine continuity and differentiability.