# Thread: Find equations that satisfy certain properties

1. ## 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).

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

2) i cant think of such a function

3. Originally Posted by tiki84626

2) k: [0,infinity) -> R is continuous, but k has no maximum and no minimum (again global min and max).
$\displaystyle k(x)=\begin{cases} 0 & \mbox{if} \quad x=0 \\ x\sin\left(\frac{1}{x}\right) & \mbox{if} \quad x>0\end{cases}$

4. ## Is k continuous?

Is the function k that you've described continuous?

5. Originally Posted by tiki84626
Is the function k that you've described continuous?
Shouldn't you figure that out on your own?

6. Originally Posted by tiki84626
Is the function k that you've described continuous?
Well, clearly $\displaystyle xsin(\frac{1}{x})$ is defined for $\displaystyle x > 0$. Now all you have to do is check that the limit of $\displaystyle xsin(\frac{1}{x})$ as $\displaystyle x \rightarrow 0$ is $\displaystyle 0$ (which it is).