# continuous

• Nov 17th 2010, 08:39 AM
alexandrabel90
continuous
how do you prove that the unction f(x) = tan(x+x^2)/(1+x+x^2) is continuous for all x=[0,1] except at one point.

i found out that it is not cont at x= 0.84936..but i dont know how to prove that it is continuous now..
• Nov 17th 2010, 09:53 AM
chisigma
The function $f(\theta) = \tan \theta$ has a singularity for $\theta= \frac{\pi}{2}$ so that Your function has a singularity for x satisfying the equation $x^{2}+x-\frac{\pi}{2}=0$. One solution is $\displaystyle x= \frac{-1 + \sqrt{1+2 \pi}}{2} = .849368862...$

Kind regards

$\chi$ $\sigma$
• Nov 17th 2010, 09:59 AM
alexandrabel90
haha ya i knew that it is not continuous at x= 0.84936.. as stated in my question..so, im trying to ask how do i show that it is continuous for all points other than x= 0.84936..
• Nov 17th 2010, 10:00 AM
FernandoRevilla
The function:

$f(x)=\dfrac{\tan (x+x^2)}{1+x+x^2}$

is an elementary function, $1+x+x^2\neq 0$ for all $x$ real and $\tan \pi/2$ does not exists. Solving $x^2+x=\pi/2$ you'll obtain the point in $[0,1]$ where $f$ is not continuous:

$x=\dfrac{-1+\;\sqrt[]{1+2\pi}}{2}\in{[0,1]}$

Regards.

Edited: Sorry, I didn't see the previous answers.
• Nov 17th 2010, 10:09 AM
FernandoRevilla
Quote:

Originally Posted by alexandrabel90
... so, im trying to ask how do i show that it is continuous for all points other than x= 0.84936..

Theorem: If f is an elementary real funtion of real variable, then f is continuos just at the points f is defined.

Regards.
• Nov 17th 2010, 10:19 AM
alexandrabel90
may i know how do i prove this theorem? i never learnt this theorem before. thanks!!
• Nov 17th 2010, 10:47 AM
FernandoRevilla
For example:

(i) $f_1(x)=1+x+x^2$ is continuous in $[0,1]$ (polynomical function).

(ii) $f_2(x)=x+x^2$ is continuous in $[0,1]$ (polynomical function).

(iii) $f_3(x)=\tan x$ is continuous in $\ldots \cup (-\pi/2,\pi/2)\cup\ldots$

(iv) Composition of continuous are continuous.

etc. etc., ...

Regards.