Function undefined at non-integer values [closed]

Question

Is there a function $f(x)$ which is not defined at integer values?

Please do NOT answer $f(x) = \begin{cases} a, & \text{if } x \in \mathbb{Z}, \\ \text{undefined}, & \text{otherwise} \end{cases}$

I thought about $f(x)=x!$ but it turns out $f(x)=\Gamma(x+1)$

So any ideas? Thanks a lot!

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P.S. - I guess $\sqrt{-\{x\}}$ is a solution!

Answer 1

$$ f(x)=\frac{1}{x-\lfloor{x}\rfloor} $$

The function is defined for all x, except for integers!

Answer 2

For example

$$f(x) = \frac{1}{\lfloor\frac{1}{2}(\cos(2 \pi x) + 1) \rfloor}$$

if $x \in \mathbb{Z}$

$$f(x) = \frac{1}{\lfloor\frac{1}{2}(1 + 1) \rfloor} = \frac{1}{\lfloor 1 \rfloor} = 1$$

if $x \not \in \mathbb{Z}$

$$-1 \leq \cos(2 \pi x) < 1$$ $$0 \leq \frac{1}{2}(\cos(2 \pi x) + 1) < 1$$

So $$\lfloor\frac{1}{2}(\cos(2 \pi x) + 1) \rfloor = 0$$

And we get $$f(x) = \frac{1}{0} $$

which is undefined.

Another function is

$$f(x) = \sqrt{\cos(2 \pi x) - 1}$$