My math question

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I came across this in a book I am reading.

Let f(x) be equal to the nth decimal place digit of x (for our consideration lets say the 100th).

Is this a function? Is there any special name for it, or is it famous?

Is it continuous? Is it differentiable?
 

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  • #2
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f is a function on R if f(x) is defined for all real x. For numbers with two decimal representations, such as 0.999... = 1.000..., you'll have to pick one representation. So f(1) = 0 or f(1) = 9 as it can't be both.

I don't know about the function's name/fame.

The range is {0, 1, ..., 9}. Can f be continuous with this range?
 
  • #3
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Can you clarify that? Do you mean something along the lines of:

f(x) = 4th decimal place;

f(5,000) = 5 ; f(6,000) = 6 ; f(17,243) = 7 ???
 
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No I think he meant e.g. f(123.45678) = 7.
 
  • #5
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I suppose this is just a step function, continuous on some open interval. If we took the first digit, it would be the usual step function, but if we take the second digit, we are reducing the continuous length by a factor of 10--and so forth. But it is still continuous over some interval, and there it would be differential with value 0.
 
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I see now that it is not continuous, -- over all of R, -- but yes over some open intervals..

"f is a function on R if f(x) is defined for all real x. For numbers with two decimal representations, such as 0.999... = 1.000..., you'll have to pick one representation. So f(1) = 0 or f(1) = 9 as it can't be both."

Sure pick the min possible.

"No I think he meant e.g. f(123.45678) = 7."

Yes thats right sorry for the ambiguity.

"suppose this is just a step function, continuous on some open interval. If we took the first digit, it would be the usual step function, but if we take the second digit, we are reducing the continuous length by a factor of 10--and so forth. But it is still continuous over some interval, and there it would be differential with value 0."

Yes I see that now, cool, thanks!
 
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Any continuous function that takes integer values is constant on each connected component of its domain.
 
  • #8
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I don't think such a function can be well-defined. For example it is well known that 1.0000000.... = 0.9999999..... Here f(x) gives different values for the same x.

Edit: Sorry I see this was already discussed above.
 
  • #9
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I don't think such a function can be well-defined. For example it is well known that 1.0000000.... = 0.9999999..... Here f(x) gives different values for the same x.

Edit: Sorry I see this was already discussed above.
It can be well-defined, provided that you define it consistently in cases of ambiguity. The indicator function of the Cantor set is similar.
 

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