Stewart – Calculus – 2.5 – Continuity

September 16, 2009 Clifton Leave a comment

Definition: A function f is continuous at a number a if

This definition implicitly requires three things if f is continuous at a:

1. f(a) is defined (that is, a is in the domain of f)

2. $\lim_{x \to a}f(x)$ exists

3. $\lim_{x \to a}f(x)=f(a)$
If $f$ and $g$ are continuous functions with $f(3)=5$ and
$\lim_{x \to 3}[2f(x) - g(x)]=4$ , find $g(3)$ .

Use the definition of continuity and the properties of limits to show that the funtion is continous at the given number a. $f(x)=(x+2x^{3})^{4}$ , $a=-1$
The Intermediate Value Theorem
Suppose that $f$ is continuous on the closed interval $[a, b]$ and let $N$ be any number between $f(a)$ and $f(b)$ , where $f(a) \neq f(b)$ . Then there exists a number $c$ in $(a,b)$ such that $f(c)=N$ .

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Use the Intermediate Value Theorem to show that there is
a root of the given equation in the specified interval.
$x^4+x-3=0, (1,2)$

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