Question

let f be continuous on R ,let f'(x) exit for all x don't eqaul to 0, and let limit n approaches to 0 f'(x)=0, prove that f'(0) exit's. what is it's value?​

Answer 1

Whatever you have written before, the claim:

the condition of f being measurable is in the case at hand equivalent to it being of bounded variation.

is not true. That is all I wanted to remark in my last respond to you.

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Moreover, returning to your last answer,

"boundedness of variation was stated only as a sufficient condition."

I can agree that you have stated this.

NOTE HOWEVER, PLEASE, THAT:

boundedness of the variation of a differentiable function f defined on (a,\infty) is NOT sufficient for the implication:

If limit of f at infinity equals 0 then the limit of its derivative f' at infinity is also zero.

An example (which must be pretty tedious for presenting briefly in this thread) is in preparation.