Continuous functions form a unital algebra

From Calculus
Jump to: navigation, search

Statement

Continuity at a point version

Suppose c \in \R. Then, the following are true:

  • Additive closure: If f,g are functions defined in open intervals containing c and both of them are continuous at c, then the pointwise sum f + g is continuous at c.
  • Scalar multiples: If f is defined in an open interval containing c and is continuous at c, and \lambda is a real number, then \lambda f is continuous at c.
  • Multiplicative identity: The constant function that sends everything to 1 is continuous at c.
  • Multiplicative closure: If f,g are functions defined in open intervals containing c and both of them are continuous at c, then the pointwise product f \cdot g is continuous at c.

There is a technical way of forming a unital algebra (a vector space endowed with an identity and a compatible multiplication) from the set of continuous functions at the point c, once we identify (as being the same) any two functoins that agree on an open interval containing c. This approach is called taking the germ of the function.

Continuity around a point version

Suppose c \in \R. Then, the following are true:

  • Additive closure: If f,g are functions defined and continuous in open intervals containing c, then the pointwise sum f + g is continuous at c.
  • Scalar multiples: If f is defined and continuous in an open interval containing c>, and \lambda is a real number, then \lambda f is continuous at c.
  • Multiplicative identity: The constant function that sends everything to 1 is continuous in an open interval containing c.
  • Multiplicative closure: If f,g are functions defined and continuous in open intervals containing c, then the pointwise product f \cdot g is continuous at c.

There is a technical way of forming a unital algebra (a vector space endowed with an identity and a compatible multiplication) from the set of continuous functions at the point c, once we identify (as being the same) any two functoins that agree on an open interval containing c. This approach is called taking the germ of the function.

Continuity on an interval version

Suppose I is an interval (possibly open, closed, or infinite from the left side and possibly open, closed, or infinite from the right side -- so it could be of the form [a,b],[a,b),(a,b),(a,b],(-\infty,b),(-\infty,b],(a,\infty),[a,\infty),(-\infty,\infty)). A continuous function on I is a function on I that is continuous at all points on the interior of I and has the appropriate one-sided continuity at the boundary points (if they exist).

The continuous functions on I form a real vector space, in the sense that the following hold:

  • Additive: A sum of continuous functions is continuous: If f,g are both continuous functions on I, so is the pointwise sum of functions f + g.
  • Scalar multiples: If \lambda \in \R and f is a continuous function on I, then \lambda f is also a continuous function on I.
  • Multiplicative identity': The constant function sending everything to 1 is a continuous function on I.
  • Multiplicative closure: A product of continuous functions is continuous: If f,g are both continuous functions on I, so is the pointwise product of functions f \cdot g.

Facts used

  1. Continuous functions form a vector space: This takes care of the addition and scalar multiples aspects
  2. Limit is multiplicative: This takes care of multiplicative closure