TL8 Differentiability and Continuity

Core Concepts¶

Differentiability¶

Differentiability (Precise)

A function is differentiable at $x$ provided the limit exists:

f'(x)=\lim_{h\to 0}\dfrac{f(x+h)-f(x)}{h}

It is said to be non-differentiable at $x$ if this limit does not exist.

Differentiability (Intuitive)

A function is differentiable at $x$ provided its graph does not have:

Jumps, breaks, or holes
Sudden changes in slope
Vertical tangents

Continuity¶

Continuity (Precise)

A function is continuous at $x=a$ provided the limit exists:

\lim_{x\to a}f(x)=f(a)

It is said to be discontinuous at $x=a$ if this equation does not hold.

Differentiability (Intuitive)

A function is continuous at $x=a$ provided its graph does not have any:

Jumps, breaks, or holes

Relationship¶

There is a relationship between these two concepts of differentiability and continuity. Differentiability is the stronger condition: if we know a function is differentiable at $x=a$ , then automatically we know that it is also continous there as well.

The converse of this, is not alway true. We can have continuous functions that are not differentiable at point.

Primary Examples¶

Homework¶

Write out each question as well as your solution on the homework page template.

Make sure you leave a good amount of blank space between each question.
Write down the question as well as your solution.