- #1

corey115

- 8

- 0

## Homework Statement

A function

*f*:[a,b] [itex]\rightarrow[/itex] ℝ is called piecewise continuous if there exists a finite number of points

*a = x*such that

_{0}< x_{1}< x_{2}< ... < x_{k-1}< x_{k}= b(a)

*f*is continuous on (

*x*) for

_{i-1}, x_{i}*i = 0, 1, 2, ..., k*

(b) the one sided limits exist as finite numbers

Let

*V*be the set of all piecewise continuous functions on [

*a, b*]. Prove that

*V*is a vector space over ℝ, with addition and scalar multiplication defined as usual for functions.

## Homework Equations

The axioms for fields and vector spaces.

## The Attempt at a Solution

If function addition and scalar multiplication work in the same way as usual, don't I just have to make an argument that the limits still exist as finite numbers (since our scalars are finite)? And a similar argument that if I add two functions that those limits will just be the sum of the two limits of each function?