Real function/Continuous/Rules/Section

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Let and be subsets and let


denote functions with

. Then the following statements hold.
  1. If is continuous in and is continuous in , then also the composition is continuous in .
  2. If and are continuous, so is .

The first statement follows from fact. This implies also the second statement.

Let be a subset and let

be continuous functions. Then also the functions

are continuous. For a subset such that has no zero in , also

is continuous.

This follows from fact and fact.

Polynomial functions



Due to example and fact, the powers

are continuous for every . Hence, also the functions

are continuous for every and therefore, again due to fact, the functions

are continuous.

Rational functions are continuous on their domain.

Let be polynomials and let . Then the rational function

is continuous.

This follows from fact and fact.