# Stewart – Calculus – 2.5 – Continuity

1. 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. $\lim_{x \to a}f(x)=f(a)$

2. 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. $f(x)=(x+2x^{3})^{4}$, $a=-1$

3. 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 $f(c)=N$.

~ ~ ~

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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