Continuity

You can plug a in to find the limit if the function is continuous.

Limit Approaching Value Definition

${\lim }_{x\to c}f(x)=f(c)$

Definition

A function $f(x)$ is continuous at $x=a$ if ${\lim }_{x\to a}f(x)=f(a)$ This definition can be thought of as a 3 part definition:

1. $f(a)$ is defined
2. ${\lim }_{x\to a}f(x)$ exists. that is, ${\lim }_{x\to a^{-}}f(x)={\lim }_{x\to a^{+}}f(x)$
3. ${\lim }_{x\to a}f(x)=f(a)$ (the numbers from (1) and (2) are equal) We say that $f$ is discontinuous at $a$ if $f$ is not continuous at $a$

Epsilon-Delta Definition

$\forall ϵ>0,\exists \delta >0\text{ such that }|x-c|<\delta \Rightarrow |f(x)-f(c)|<ϵ$

Graphical Interpretation

Euler expressed a continuous function as “a curve described by freely leading the hand”, you can draw the graph without lifting your pencil from the paper Note that this definition is not rigorous enough

Left & Right Continuity

Left Continuous

${\lim }_{x\to a^{-}}f(x)=f(a)$

Right Continuous

${\lim }_{x\to a^{+}}f(x)=f(a)$

Continuous Rules (Derived from Limits)

• Continuous Sums
• Continuous Products
• Continuous Compositions

Other

• Continuity Theorem
• Uniform Continuity