# Always a double root between “no roots” and “at least one root”?

Let $f(x,y)$ be a real bivariate polynomial. Suppose that $f(x,.)$ has no real roots when $x<0$, but has at least one real root when $x>0$. Does it automatically follow that $f(0,.)$ has a double root ?

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No, consider $f(x,y)=1-xy^2$. 
In that counterexample, the roots go to infinity when $x\to 0$. Which leads me to wonder what happens when the roots stay in a bounded interval, I updated my initial question – Ewan Delanoy Jul 27 '14 at 14:40