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i solved the following equation and got this answer: $y'(x)=y(x)^2 , y(0)=1 \Rightarrow y(x)=\frac{1}{1-x}$

this must always have positive rate of change but at the $x=1$ it is not continious and jumps from $+\infty$ into $-\infty$, why is that?

if that be a model of population what would happen at that point?(assuming the model exist)

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    Well, at $x=1$, the function nor the derivative exist. – 2017-02-26
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    If you took the standpoint of someone looking at the Riemann sphere, you would have argued that $+\infty$, $-\infty$, and all complex infinities are the same, hence, no discontinuity at the poles. – 2017-02-26
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    Solutions of differential equations are defined on intervals containing the point at which one is given an initial condition. In your case, the initial point is $x=0$ and for the initial condition $y(0)=1$, the maximal interval of definition ihappens to be $(-\infty,1)$. For other initial conditions $y(0)$ of for other differential equations, the maximal interval could be different. You might have, until now, only met situations where the maximal interval is $(-\infty,\infty)$ for every initial condition, hence your (natural) puzzlement... – 2017-02-26

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What you are seeing is called Finite time blow-up. The solution of the differential equation reaches $+\infty$ in a finite time. The maximal interval of existence of the solution is $([0,1)$ (or $(-\infty,1)$.)

Interpreted as a population model, the equation says that the population grows proportionally to the square of the population. This growth is so fast, that the population becomes infinite in a finite time.