Let be continuous at . Then and are continuous at , and if for all , then is continuous at as well. Continuity therefore behaves like an algebra under pointwise operations, mirroring the algebra of limits.

Let , , . If is continuous at and is continuous at , then is continuous at .

Given , continuity of at supplies with for ; continuity of at then supplies with . Chaining the two implications,

so composition preserves continuity.

• is continuous: take , so .

• is not continuous at , since does not exist.

• The Dirichlet function is discontinuous at every : rationals are approached by irrational sequences with , and irrationals by rational sequences with .

• is continuous at every : for ,

by taking , so once is large enough.