# Continuity

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##### Intros

###### Lessons

__Introduction to Continuity__- Discuss CONTINUITY in everyday language: a function whose graph has no break in it.
- 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: - Lesson Overview - Continuity

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##### Examples

###### Lessons

**Discussing "Continuity" Graphically**

The graph of a function $f$ is shown below.- State the numbers at which $f$ is discontinuous.
- Explain and classify each discontinuity.

**Detecting Discontinuities**Are the following functions continuous at $x=3$?

i) $f(x)=\frac{x^2-2x-3}{x-3}$

ii)

iii)

**Discontinuities of Rational Functions (denominator=0)**

Locate and classify each discontinuity of the function: $f(x)=\frac{x^2-9}{x^2+x-6}$**Discussing "Continuity" Algebraically**

Find the values of a and b that make the function $f$ continuous on ($-\infty$, $\infty$).