Continuity of roots of a polynomial. Let us consider a polynomial  $p(x)=ax^2+bx+c$ of degree $2$ with $a,b,c$ coming from $\Bbb R$. Now as the value of constant $'a'$ changes continuously, will the root of $p(x)$ also change continuously?
I think it will. If someone can prove or disprove it.
Edit:Can someone give some intutive proof for a general n degree polynomial?

If we are considering complex roots that is true. Loosely speaking, if the distance between $(a,b,c)$ and $(a',b',c')$ is small (say, $\leq \varepsilon$) by evaluating $a'x^2+b'x+c'$ at the roots of $ax^2+bx+c$ we get something small ($\leq K\varepsilon$), so the roots of $a'x^2+b'x+c'$ cannot be too far from the roots of $ax^2+bx+c$. Indeed this argument is simple to turn into a rigorous one (by the inverse function theorem) if the roots of $ax^2+bx+c$ are simple (no pun intended). If there are multiple roots (that happens when the discriminant is zero) a bit of extra care is needed. For instance, if for a quadratic real polynomial the term $b^2-4ac$ goes from being positive to being negative, the roots go from being real to being complex.
