Continuous functions form a vector space

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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.

There is a technical way of thinking of the collection of such functions as a real vector space, that basically involves identifying two functions as being the same function if they agree on an open interval containing c. This idea of identifying functions that look the same around c is called taking the germ of a function and is beyond the scope of single variable calculus.

Continuity around 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 on open intervals containing c, then the pointwise sum f + g is continuous on an open interval containing c.
  • Scalar multiples: If f is defined and continuous in an open interval containing c and is continuous at c, and \lambda is a real number, then \lambda f is continuous on an open interval containing c.

There is a technical way of thinking of the collection of such functions as a real vector space, that basically involves identifying two functions as being the same function if they agree on an open interval containing c. This idea of identifying functions that look the same around c is called taking the germ of a function and is beyond the scope of single variable calculus.

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 closure: A sum of continuous functions is continuous: If f,g are both continuous functions on I, so is 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.

We can also frame this in terms of linear combinations: if f_1,f_2,\dots,f_n are all continuous functions and a_1,a_2,\dots,a_n \in \R, then the function

x \mapsto a_1f_1(x) + a_2f_2(x) + \dots + a_nf_n(x)

is also a continuous function.

Facts used

  1. Limit is linear: This says that the limit of the sum is the sum of the limits, and scalar multiples can be pulled out of limits.