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For i) I said for a in [0,1], then the group of units are = {f in R | f(a) =/= 0}

i.e a continous function f on [0,1] would have a continous function g on [0,1] such that f.g=1

but the function would have to be g = 1/f, but this wouldnt be continous if f(a) = 0

for ii) I have to show for a,b in R then if ab=0 then either a = 0 or b = 0

but I havn't gotton any further here since I'm having a hard time finding functions a and b where the product is 0

for iii) I believe I have to construct a surjective ring homomorphism ( f: R -> F ) and show that the kernal is equal to the Ideal.

But I'm not sure how what field F to choose for this

Or I have to show that the quotient group R/I is a field.

Thanks for any help!

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