Continuity of rational functions between affine algebraic sets
Solution 1:
A polynomial is continuous by definition of the Zariski topology. A rational function is a function that can be written locally (i.e. on an open neighborhood of each point) as a quotient $\frac{f}{g}$ where $f$ and $g$ are polynomial such that $g$ never vanishes on the open neighborhood. As a quotient of such (continuous) polynomials is continuous, you see that a rational function is locally continuous. As the notion of continuity is purely local on the source (which is a fancy way to say that "locally continuous" implies "continuous") it is also continuous.