Continuous functions form a vector space

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 multiplies: 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.