Is mod x differentiable at all points

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Discussion Overview

The discussion centers on the differentiability of the function f(x) = |x|, commonly referred to as the absolute value function, particularly at the point x = 0. Participants explore the implications of using the term "mod" in this context and clarify the function's behavior across its domain.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants question the use of "mod" to refer to the absolute value function, suggesting that "mod" typically denotes a modulus operation in modular arithmetic.
  • Others clarify that "modulus" is indeed another term for absolute value, indicating a potential misunderstanding in terminology.
  • It is proposed that f(x) = |x| can be expressed as a piecewise function, which is differentiable in the intervals (0, ∞) and (-∞, 0), with the only concern being differentiability at x = 0.
  • One participant suggests that proving non-differentiability at x = 0 is straightforward, although the specifics of this proof are not detailed.

Areas of Agreement / Disagreement

There is no consensus on the terminology used to describe the absolute value function, with some participants expressing confusion over the term "mod." However, there is agreement that the function is differentiable at all points except x = 0, though the nature of its differentiability at that point remains a topic of discussion.

Contextual Notes

Participants have not fully resolved the implications of using different terminologies for the absolute value function, and the discussion includes varying interpretations of the function's differentiability at x = 0.

sphyics
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f(x)= mod x is this function differentiable at all points other than 0.
 
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What do you mean by f(x) = mod x?
Usually, mod is used in an expression like "6 mod 4 = 2", and "mod 4" by itself doesn't really mean anything.

Did you mean "f(x) = |x|", the absolute value of x?
In that case, the answer is yes: you can easily see this by showing that it is equivalent to x for x > 0 and -x for x < 0. To prove that it is non-differentiable at x = 0, you need to do a little work, but that's still straightforward.
 
CompuChip said:
Did you mean "f(x) = |x|", the absolute value of x?
In that case, the answer is yes: you can easily see this by showing that it is equivalent to x for x > 0 and -x for x < 0. To prove that it is non-differentiable at x = 0, you need to do a little work, but that's still straightforward.
yes its absolute value of x :)
 
Why did you use the term mod for absolute value? I'm not familiar with that application of mod.
 
I think the OP meant modulus, which is another term for absolute value.
 
gb7nash said:
I think the OP meant modulus, which is another term for absolute value.

Don't use unapproved abreviations! Haha, i got confused.
 
Yes; one way of seeing it is look at |x| as the piecewise function:

f(x)=x , if x≥0
f(x)=-x, if x≤0

Then the function is linear , and if you accept that for n>1, d/dx(xn)=nxn-1, then f is differentiable in (0,∞), and in (-∞,0), and the only possible problem is at x=0.
 

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