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## Main Question or Discussion Point

Hi PF!

I want to make sure I understand the notion of a subspace. Our professor gave an example of one: the set of degree n polynomials is a subspace of continuous functions. This is because a) a polynomial is intrinsically a subset of continuous functions and b) summing any polynomials yields another polynomial (closed under addition) and c) any polynomial multiplied by a constant is still a polynomial (closed under scalar multiplication).

If my reasoning is correct, then I think the set of discontinuous functions is NOT a subset of all real functions. To see why, consider ##f(x) = 1## everywhere except non-existent at ##x=1##. Then take the function ##g(x)=1## when ##x=1## and non-existent everywhere else. Then the sum is clearly continuous. (How would this change though if I changed non-existent to zero? I know the result is the same, but is ##g(x)## as I have defined it even discontinuous--I realize this is a real analysis question.)

I want to make sure I understand the notion of a subspace. Our professor gave an example of one: the set of degree n polynomials is a subspace of continuous functions. This is because a) a polynomial is intrinsically a subset of continuous functions and b) summing any polynomials yields another polynomial (closed under addition) and c) any polynomial multiplied by a constant is still a polynomial (closed under scalar multiplication).

If my reasoning is correct, then I think the set of discontinuous functions is NOT a subset of all real functions. To see why, consider ##f(x) = 1## everywhere except non-existent at ##x=1##. Then take the function ##g(x)=1## when ##x=1## and non-existent everywhere else. Then the sum is clearly continuous. (How would this change though if I changed non-existent to zero? I know the result is the same, but is ##g(x)## as I have defined it even discontinuous--I realize this is a real analysis question.)