Definition of Continuous function.

• Jul 11th 2011, 01:29 PM
Lamalif
Definition of Continuous function.
Hi all member.
Can any one help me to demonstrat that (by definition ):
does the folowing function is continuous at x=2.
f(x)=1/(1+x^4).
Thanks.
• Jul 11th 2011, 02:04 PM
pickslides
Re: Definition of Continuous function.
Will be continuous if

1. f(x) exists for x=2

2. $\displaystyle \displaystyle \lim_{x\to 2}f(x)$ must also exist

3. f(2) = $\displaystyle \displaystyle \lim_{x\to 2}f(x)$
• Jul 11th 2011, 02:13 PM
Plato
Re: Definition of Continuous function.
Quote:

Originally Posted by Lamalif
Hi all member.
Can any one help me to demonstrat that (by definition ):
does the folowing function is continuous at x=2.
f(x)=1/(1+x^4).

If you are looking for a $\displaystyle \epsilon-\delta$ proof the you need
$\displaystyle \left| {\frac{1}{{1 + x^4 }} - \frac{1}{{17}}} \right| = \frac{{\left| {16 - x^4 } \right|}}{{17\left( {1 + x^4 } \right)}} \leqslant \left| {2 - x} \right|\left| {2 + x} \right|\left| {4 + x^2 } \right|$

You can control the size of $\displaystyle |x-2|$ factor and bound the other two.