# Continuity

##### Intros
###### Lessons
1. Introduction to Continuity
2. Discuss CONTINUITY in everyday language: a function whose graph has no break in it.
3. Discuss CONTINUITY in the context of Calculus:
A function $f$ is continuous at a number a, if: $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)$

Classify different types of discontinuities: removable VS. infinite VS. jump

Analyze: rational function with a hole: $f(x)=\frac{(x-1)(x-2)}{(x-2)}$
VS.
rational function with an asymptote: $g(x)=\frac{1}{x+1}$
VS.
piecewise function:
4. Lesson Overview - Continuity
##### Examples
###### Lessons
1. Discussing "Continuity" Graphically
The graph of a function $f$ is shown below.
1. State the numbers at which $f$ is discontinuous.
2. Explain and classify each discontinuity.

2. Detecting Discontinuities Are the following functions continuous at $x=3$?
i) $f(x)=\frac{x^2-2x-3}{x-3}$

ii)
iii)
1. Discontinuities of Rational Functions (denominator=0)
Locate and classify each discontinuity of the function: $f(x)=\frac{x^2-9}{x^2+x-6}$
1. Discussing "Continuity" Algebraically
Find the values of a and b that make the function $f$ continuous on ($-\infty$, $\infty$).