Piece-wise continuous functions on the close interval

1. Aug 8, 2005

physicsss

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.

2. Aug 8, 2005

hypermorphism

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.

3. Aug 9, 2005

physicsss

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

4. Aug 9, 2005

matt grime

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.

5. Aug 9, 2005

mathwonk

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