Let F be the ring of all functions from R to R w.r.t. pointwise, addition and
multiplication. Let S be the set of all differentiable functions in F. Check whether S
is
i) a subring of F,
ii) an ideal of F.
i) Let us check whether is a subring of . Let that is are differentiable functions. Then the function is also differentiable and and hence Also the function is differentiable and and hence Therefore, we conclude that is a subring of
ii) Let us check whether is an ideal of . Let for example Taking into account that the constant function is differentiable, that is but we conclude that is not an ideal of .
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