# Piece-wise continuous functions on the close interval

## Main Question or Discussion Point

Let D[a,b] be the set of piece-wise continuous functions on the close interval [a,b]. Show that D[a,b] is a subspace of the vector space P[a,b] of all functions defined on the interval [a,b].

Can someone get me started? Do I just need to show that they are closed under addition and subtraction? If so, how do I show that? Thanks.

## Answers and Replies

Related Linear and Abstract Algebra News on Phys.org
Don't forget to show that the zero function is piecewise continuous. Just go over the definitions of a vector space and piecewise continuous function.

How do I make up and write out two piecewise functions and do operations on them?

matt grime
Science Advisor
Homework Helper
Let f and g be piecewise and continuous (write out the definition, soemthing like, for f there is a partition of [a,b] into finitely many subintervals of nonero length such that f is continuous on each, same for g) then check that f+g sastisfies this definition (which may requiire you to think for a second to porduce the subintervals). i assume you're happy that the sum of continuous functions is continuous.

mathwonk
Science Advisor
Homework Helper
learn the definition of piecewise continuous carefully as it is subtle. (are the smaller intervals open or closed or either?) then subdivide.