Piece-wise continuous functions on the close interval

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  • #1
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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

  • #2
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
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How do I make up and write out two piecewise functions and do operations on them?
 
  • #4
matt grime
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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
mathwonk
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learn the definition of piecewise continuous carefully as it is subtle. (are the smaller intervals open or closed or either?) then subdivide.
 

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