Show that f is continious

M

mreiki

New member
Joined
Oct 8, 2006
Messages
9
Hullo ... me again
I've got this function:
Code: 
       (xcos(1/x^2) , x is a IRRATIONAL number (R\Q)
f(x) = (
       (xsin(x^2+2) , x is a RATIONAL  number (Q)
Show that f is continious in x= 0

I was wondering, do you think it is enough to show that both of the functions have the limit 0, when x is heading towards 0, and f(0) = 0, and therefore it is continious in x=0 ?

Sorry if my english is wierd, I'm from Iceland =)
 
That function makes absolutely no sense!
Could it be:
\(\displaystyle \L
f(x) = \left\{ \begin{array}{l}
x\cos (1/x)\quad x \in Q \\
x\sin \left( {x^2 + 2} \right)\quad x \in \Re \backslash Q \\
\end{array} \right.\)
 
There is no meaning for "complicated number" in English.
Please give us a definition.
 
Erm, sorry for the third time, but my dictionary ditn't have the word...
i'm talking about irrational numbers

R\Q You know. .. like sqrt(3)
 
\(\displaystyle f(x) = \left\{ \begin{array}{l}
x\cos (1/x)\quad x \in \Re \backslash Q \\
x\sin \left( {x^2 + 2} \right)\quad x \in Q \\
\end{array} \right.\)

In English Q is the set of rational numbers and \(\displaystyle \Re \backslash Q\) is the set of irrational numbers.
 
exactly ... the function is defined so that it is xcos(1/x2) when x is irrational, and xsin(x^2 + 2) when x is rational ... and the objective is to show that the function is continious in x=0
 
In any neighborhood of 0 there are both rational and irrational numbers.
So in \(\displaystyle \varepsilon > 0\quad \Rightarrow \quad \left( { - \varepsilon ,\varepsilon } \right)\) we have both rational and irrational.
 
On the contrary it is continuous.
Because |cos(x)|<1, |x||cos(x)|<|x|.
The sin(x)~0 if x~0.
 
Which is what i said in the beginning ( i just didn't show the calculations)

The original question was though if you thought that it were enough to show that both of the functions (xsin... and xcos...) have the limit 0, and that f(0) = 0.

But i'll just take that as a yes :) ( i had doubt because my teacher had said something about not comparing the functions straight ahead, and just wanted someone elses opinion)
 
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