show f not continous on a

• Nov 30th 2008, 06:55 AM
show f not continous on a
For x exsits in real numbers, there is unique n exsists in integers s.t n<=x<n+1. denote this n as [x]. Thus, we obtain a function F: real numbers arrow real numbers, x arrow [x]. e.g. [-2.5]= -3.

Let a exsist in Integers show f is not continuous at a. ( use suitable sequences which converge to a )??????

Hope this makes sense thanks.
• Nov 30th 2008, 07:02 AM
TheEmptySet
Quote:

For x exsits in real numbers, there is unique n exsists in integers s.t n<=x<n+1. denote this n as [x]. Thus, we obtain a function F: real numbers arrow real numbers, x arrow [x]. e.g. [-2.5]= -3.

Let a exsist in Integers show f is not continuous at a. ( use suitable sequences which converge to a )??????

Hope this makes sense thanks.

Condiser the sequnces for $\displaystyle a \in \mathbb{Z}$ $\displaystyle x_n=a-\frac{1}{n}$ and $\displaystyle y_n=a+\frac{1}{n}$ ;$\displaystyle n=1,2,3...$

what would $\displaystyle f(x_n)$ equal and $\displaystyle f(y_n)$

I hope this helps.

Good luck.
• Nov 30th 2008, 07:20 AM
Quote:

Originally Posted by TheEmptySet
Condiser the sequnces for $\displaystyle a \in \mathbb{Z}$ $\displaystyle x_n=a-\frac{1}{n}$ and $\displaystyle y_n=a+\frac{1}{n}$ ;$\displaystyle n=1,2,3...$

what would $\displaystyle f(x_n)$ equal and $\displaystyle f(y_n)$

I hope this helps.

Good luck.

em.. im sorry any chance a bit more in depth here?
• Nov 30th 2008, 07:36 AM
surely as n gets infinitely large $\displaystyle x_n=a-\frac{1}{n}$ tends to $\displaystyle a$
$\displaystyle y_n=a+\frac{1}{n}$ tends to $\displaystyle a$

as a result $\displaystyle f(x_n)=a-1$ ???
$\displaystyle f(y_n)=a-1$
• Nov 30th 2008, 08:38 AM
TheEmptySet
Quote:

surely as n gets infinitely large $\displaystyle x_n=a-\frac{1}{n}$ tends to $\displaystyle a$
$\displaystyle y_n=a+\frac{1}{n}$ tends to $\displaystyle a$

as a result $\displaystyle f(x_n)=a-1$ ???
$\displaystyle f(y_n)=a-1$

note that $\displaystyle x_n < a$ for all n and

$\displaystyle y_n > a$ for all n

so $\displaystyle f(x_n) = a-1$ for all n

$\displaystyle f(a)=a$

$\displaystyle f(y_n) =a$ for all n

so both sequences converge to the same value, but their immages go to two different values.

Therefore the function is not continous at a when a is an integer.
• Nov 30th 2008, 08:43 AM
$\displaystyle f(y_n) =a$ for all n