# example of a continuous funtion where the sup is not achieved

• Apr 7th 2010, 05:25 AM
alice8675309
example of a continuous funtion where the sup is not achieved
Also, if we let a and b be real number with a<b then what is an example of a bounded f on [a,b] for which sup_[a,b] f is not achieved?

What would be an example of a bounded, continuous f on [a,$\displaystyle \infty$) for which sup_[a,$\displaystyle \infty$) f is not achieved?

and what would be an example of a bounded, continuous f on [a,b) for which sup_[a,b) f is not achieved?

For the first one would it be : f(x)= { x, a<=x<=b
0, x=b}
second, f(x)= -1/x+1

third, f(x)=x ?
• Apr 7th 2010, 05:57 AM
Defunkt
For the first one you need a strong inequality sign with b (f(x) = x for a<=x<b); for the second, what happens if a = -1? Only a slight change is required..
Third is correct.
• Apr 7th 2010, 06:04 AM
alice8675309
Quote:

Originally Posted by Defunkt
For the first one you need a strong inequality sign with b (f(x) = x for a<=x<b); for the second, what happens if a = -1? Only a slight change is required..
Third is correct.

hm, would i use absolute value signs?
• Apr 7th 2010, 07:04 AM
Defunkt
Quote:

Originally Posted by alice8675309
hm, would i use absolute value signs?

That works. You can also use $\displaystyle \frac{-1}{x-a-1}$.