# Thread: function continuous at a point

1. ## function continuous at a point

f:R-->R
f(x)=1 if $x = 0$
f(x)=0 otherwise.
(a) Is f continuous at the point a = 0?
(b) Is f continuous at the point a = 1?

have i done them correctly?

Continuous at a=0 because $\lim_{x\to0}f(x)=1=f(0)$ correct?

also continuous at a=1 because $\lim_{x\to1}f(x)=0=f(1)$

thanks

2. Originally Posted by charikaar
f:R-->R
f(x)=1 if $x = 0$
f(x)=0 otherwise.
(a) Is f continuous at the point a = 0?
(b) Is f continuous at the point a = 1?

have i done them correctly?

Continuous at a=0 because $\lim_{x\to0}f(x)=1=f(0)$ correct?

also continuous at a=1 because $\lim_{x\to1}f(x)=0=f(1)$

thanks
Are you really sure of the first case? When you are not at 0 (which you aren't when you are taking a limit), the function takes the value 0. What does $x_n=0$ converge to?

I mean you can think about it more intuitively, is the function "connected" at 0? Draw it.

3. I see what you mean. The function is not connected at zero and hence the limit in first case doesn't exist so not continuous at a=0

4. Originally Posted by charikaar
I see what you mean. The function is not connected at zero and hence the limit in first case doesn't exist so not continuous at a=0
The limit does exist! Try to compute it this way
$
f(1/2), f(1/4), f(1/8),...
$

what it the limit of this sequence? (it's the same as $\lim_{x \downarrow 0} f(x)$).

I think it may help if you look up the definition of a limit. The limit at a=0 might not be f(a) (which is the case here), but indeed it still does exist.