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All our movies available on the website are free to download and watch without any registration.Q:
A simple continuous probability density function and a singularity at $0$
I am trying to find a function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that:
$f(0)=1$
$f(x)>0$ for every $x\in\mathbb{R}\setminus \{0\}$
$f(x)$ is continuous
But i could not find an answer. In the case that we have a probability density function $\rho(x)$ such that $\rho(x)>0$ then we can define $f(x)=\frac{\rho(x)}{\int_{ -\infty}^{+\infty}\rho(x)dx}$. In this case we have a continuous function that has a global maximum at $0$, hence the function will be undefined at every $x
eq 0$ (as $\rho(x)$ is undefined). I wonder if there is a similar simple function for this case, or whether it is possible to find a function $f$ such that is continuous at every $x
eq 0$?
Thanks in advance
A:
As noted by Ron, $f(x)=1$ if $x>0$ and $f(x)=-1$ if $x
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