Piece-wise continuous functions on the close interval

[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.

 

[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.

 

[physicsss]

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

 

[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.

 

[mathwonk]

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