1-8: Continuity

1. Definition A function $\displaystyle f$ is continuous at a number $\displaystyle a$ if $\displaystyle \lim_{x\rightarrow a}f(x)=f(a)$
Notice that Definition 1 implicitly requires three things if $\displaystyle f$ is continuous at $\displaystyle a$
1.$\displaystyle f(a)$ is defined (that is, $\displaystyle a$ is in the domain of $\displaystyle f$ )
2$\displaystyle \lim_{x\rightarrow a}f(x)$ exists
3$\displaystyle \lim_{x\rightarrow a}f(x)=f(a)$

2. Definition A function $\displaystyle f$ is continuous from the right at a number $\displaystyle a$ if
$\displaystyle \lim_{x\rightarrow a^{+}}f(x)=f(a)$
and $\displaystyle f$ is continuous from the left at $\displaystyle a$ if
$\displaystyle \lim_{x\rightarrow a^{-}}f(x)=f(a)$

3. Definition A function $\displaystyle f$ is continuous on an interval if it is continuous at every number in the interval. (if $\displaystyle f$ is defined only on one side of an endpoint of the interval, we understand continuous at the endpoint to mean continuous from the right or continuous from the left)

4. theorem If $\displaystyle f$ and $\displaystyle g$ are continuous at $\displaystyle a$ and $\displaystyle c$ is a constant, then the following functions are also continuous at $\displaystyle a$:
1$\displaystyle f+g$
2$\displaystyle f-g$
3$\displaystyle cf$
4$\displaystyle fg$
5$\displaystyle \frac{f}{g}$ if $\displaystyle g(x)\neq 0$

5. theorem

(a) Any polynomial is continuous everywhere; that is, it is continuous on  $\displaystyle \mathbb{R}=(-\infty ,\infty )$

(b) Any rational function is continuous wherever it is defined; that is, it is continuous on its domain

6$\displaystyle \lim_{\theta \rightarrow 0}cos \theta =1$   $\displaystyle \lim_{\theta \rightarrow 0}sin \theta =0$

7. theorem The following types of functions are continuous at every number in their domains:

polynomials    rational functions

root functions    trigonometric functions

8. theorem If $\displaystyle f$ is continuous at $\displaystyle b$ and $\displaystyle \lim_{x\rightarrow a}g(x)=b$ then $\displaystyle \lim_{x\rightarrow a}f(g(x))=f(b)$ In other words,
$\displaystyle \lim_{x\rightarrow a}f(g(x))=f(\lim_{x\rightarrow a}g(x))$

9. theorem If $\displaystyle g$ is continuous at $\displaystyle a$ and $\displaystyle f$ is continuous at $\displaystyle g(a)$, then the composite function $\displaystyle f\circ g$ given by $\displaystyle (f\circ g)(x)=f(g(x))$ is continuous at $\displaystyle a$

10. The Intermediate Value Theorem Suppose that $\displaystyle f$ is continuous on the closed interval $\displaystyle \left [ a,b \right ]$ and let $\displaystyle N$ be any number between $\displaystyle f(a)$ and $\displaystyle f(b)$, where $\displaystyle f(a)\neq f(b)$
Then there exists a number $\displaystyle c$ in $\displaystyle (a,b)$ such that $\displaystyle f(c)=N$

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