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why arent non continuous functions in an interval a linear space?

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- #1

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why arent non continuous functions in an interval a linear space?

- #2

CompuChip

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Hint: try looking at closure under addition: can you find two discontinuous functions f and g such that f + g is continuous?

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- #4

Deveno

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f(x) = -1, for 0 ≤ x < 1/2

f(x) = 1, for 1/2 ≤ x ≤ 1.

clearly, f is discontinuous (at 1/2).

now define g:[0,1]→R by:

g(x) = 1 for 0 ≤ x < 1/2

gx) = -1, for 1/2 ≤ x ≤ 1.

again, g(x) is discontinuous (at 1/2).

but (f+g)(x) = 0, for all x in [0,1], and constant functions are continuous.

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HallsofIvy

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- #6

CompuChip

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Because you wanted to show that the space ofwhy does f+g have to be continuous?

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- #8

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If V would be the set of discontinuous functions, then the above becomes: if x and y are discontinuous functions, so is x+y.

The above example shows that this is false, hence the discontinuous functions do not form a linear space.

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thanks

- #10

HallsofIvy

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