# Doing a problem on rings from Dummit & Foote I think I'm mis-reading it

## Homework Statement

Decide which of the following are subrings of the ring of all functions from the closed interval [0,1] to R (the reals)

a) The set of all functions f(x) such that f(q) = 0 for all q in Q (the rationals) & q in [0, 1]
b) The set of all polynomial functions
c) The set of all functions which have only a finite number of zeros, together with the zero function
d) The set of all functions which have an infinite number of zeros.
e) The set of all functions f such that lim {x -> 1+} f(x) = 0
f) The set of all rational linear combinations of the functions sin(nx) and cos(mx) where m and n are non-negative integers

## The Attempt at a Solution

The first one is pretty straight forward to show that it is a subring. But unless I'm mistaken, the sets mentioned from b to f aren't even subsets let alone subgroups or subrings since they can be defined on a larger domain than [0,1]. Am I correct? Or am I reading it wrong?

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jbunniii
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The first one is pretty straight forward to show that it is a subring. But unless I'm mistaken, the sets mentioned from b to f aren't even subsets let alone subgroups or subrings since they can be defined on a larger domain than [0,1]. Am I correct? Or am I reading it wrong?
I'm sure they mean that you should assume in all six parts that the functions are from [0,1] to $\mathbb{R}$, except that doesn't make sense for part (e). Does it really say

$$\lim_{x \rightarrow 1^+}f(x)$$?

Because that limit doesn't make sense if the function's domain is [0,1].

: I just checked using the "look inside" feature on Amazon; in that edition (3rd) it reads

$$\lim_{x \rightarrow 1^-}f(x)$$

which makes more sense.

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I'm sure they mean that you should assume in all six parts that the functions are from [0,1] to $\mathbb{R}$, except that doesn't make sense for part (e). Does it really say

$$\lim_{x \rightarrow 1+}f(x)$$?

Because that limit doesn't make sense if the function's domain is [0,1].
I'll take your word for it since that's the only conclusion I could come to myself.

But as for part (e), that is correct. You can search for the question inside the only online version I could find (page 231 in that edition):

https://www.amazon.com/dp/0471433349/?tag=pfamazon01-20

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jbunniii
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I'll take your word for it since that's the only conclusion I could come to myself.

But as for part (e), that is correct. You can search for the question inside the only online version I could find (page 231 in that edition):

https://www.amazon.com/dp/0471433349/?tag=pfamazon01-20
Ha, you read my mind (see my edit above)! But I'm looking at 6(e) on page 231 in the Amazon viewer right now, even zooming in to make sure. That's a minus sign, not a plus sign.

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Ha, you read my mind (see my edit above)! But I'm looking at 6(e) on page 231 in the Amazon viewer right now, even zooming in to make sure. That's a minus sign, not a plus sign.
Ah crap, you're right. I read it wrong. Well, in my edition it is + not -.. I should probably get a newer edition :(

jbunniii