# Continuity and Differentability

### Continuity:

• Definition: A function $f\left(x\right)$ is continuous at a point $c$ if three 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 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.
• $f\left(g\left(x\right)\right)$ is continuous at $x=c$ if both $f\left(x\right)$ and $g\left(x\right)$ are continuous at $x=c$.

### Differentiability:

• Definition: A function $f\left(x\right)$ is differentiable at a point $c$ if the derivative ${f}^{\mathrm{\prime }}\left(c\right)$ exists.

• The derivative represents the rate of change of a function at a given point.
• Conditions for Differentiability:

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

• Sum/Difference Rule: $\left(f±g{\right)}^{\mathrm{\prime }}\left(x\right)={f}^{\mathrm{\prime }}\left(x\right)±{g}^{\mathrm{\prime }}\left(x\right)$
• Product Rule: $\left(f\cdot g{\right)}^{\mathrm{\prime }}\left(x\right)={f}^{\mathrm{\prime }}\left(x\right)\cdot g\left(x\right)+f\left(x\right)\cdot {g}^{\mathrm{\prime }}\left(x\right)$
• Quotient Rule: ${\left(\frac{f}{g}\right)}^{\mathrm{\prime }}\left(x\right)=\frac{{f}^{\mathrm{\prime }}\left(x\right)\cdot g\left(x\right)-f\left(x\right)\cdot {g}^{\mathrm{\prime }}\left(x\right)}{\left(g\left(x\right){\right)}^{2}}$
• Chain Rule: $\left(f\left(g\left(x\right)\right){\right)}^{\mathrm{\prime }}={f}^{\mathrm{\prime }}\left(g\left(x\right)\right)\cdot {g}^{\mathrm{\prime }}\left(x\right)$
• 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.