Are rational functions between affine algebraic sets continuous with respect

EnvivyEvoxys6

EnvivyEvoxys6

Answered question

2022-07-10

Are rational functions between affine algebraic sets continuous with respect to the Zariski topology on their domain of definition?

Answer & Explanation

lywiau63

lywiau63

Beginner2022-07-11Added 13 answers

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 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.

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