# Find equations that satisfy certain properties

• February 3rd 2010, 03:37 PM
tiki84626
Find equations that satisfy certain properties
Can anybody think of equations that satisfy the following properties:

1) f: (0,infinity) -> R is continuous and bounded, but f has no maximum and no minimum (that is global min and max).

2) k: [0,infinity) -> R is continuous, but k has no maximum and no minimum (again global min and max).
• February 3rd 2010, 05:40 PM
mabruka

$f(x)= arctan(x)$ restricted to your domain $(0,\infty)$
?

2) i cant think of such a function
• February 3rd 2010, 05:45 PM
Drexel28
Quote:

Originally Posted by tiki84626

2) k: [0,infinity) -> R is continuous, but k has no maximum and no minimum (again global min and max).

$k(x)=\begin{cases} 0 & \mbox{if} \quad x=0 \\ x\sin\left(\frac{1}{x}\right) & \mbox{if} \quad x>0\end{cases}$
• February 3rd 2010, 06:38 PM
tiki84626
Is k continuous?
Is the function k that you've described continuous?
• February 3rd 2010, 06:43 PM
Drexel28
Quote:

Originally Posted by tiki84626
Is the function k that you've described continuous?

Shouldn't you figure that out on your own?
• February 3rd 2010, 07:08 PM
Pinkk
Quote:

Originally Posted by tiki84626
Is the function k that you've described continuous?

Well, clearly $xsin(\frac{1}{x})$ is defined for $x > 0$. Now all you have to do is check that the limit of $xsin(\frac{1}{x})$ as $x \rightarrow 0$ is $0$ (which it is).