#Differentiability & Continuity

#Table of Contents

1. Differentiability
2. Relationship Between Differentiability and Continuity
3. Showing Non-Differentiability
4. Points of Non-Differentiability on Graphs
5. Worked Example
6. Practice Questions
7. Glossary
8. Summary and Key Takeaways

#Differentiability

#When is a Function Differentiable?

• The derivative of a function is defined as $${f}^{\prime}(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h}$$

• The derivative only exists if this limit exists.

• The derivative, if it exists, is itself a function.

• If a point is not in the domain of $f(x)$, it cannot be in the domain of $f^{\prime}(x)$.

  For instance, $f(x)=\frac{1}{x}$ is not differentiable at $x=0$ due to the vertical asymptote.

• A differentiable function has its derivative exist at each point in its domain. A function with points at which it is not differentiable can still be made a differentiable function by appropriately restricting its domain.

#Relationship Between Differentiability and Continuity

• A function must be continuous at a point to be differentiable at that point.

• However, a function being continuous at a point does not necessarily mean it is differentiable at that point.

  For instance, $f(x) = |x|$ is continuous at $x=0$, but it is not d...