Like the title suggests. Is it possible to have an implicit function that is continuous but not differentiable? Something which resembles a fractal, or is perhaps constant (not asymptotic) after a certain x but without a smooth approach, describable implicitly in x and y. On a related note can one describe a self-similar function, like a fractal, implicitly? For example a sinusoidal with noise is often self similar and always continuous but not differentiable anywhere. I am not referring to solutions given by the Implicit Function theorem which maps relations to functions.
Are there any implicit, continuous, non-differentiable functions?
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calculus
functions
implicit-differentiation
1 Answers
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Consider $f:\mathbb{R}^{2}\rightarrow \mathbb{R}$ defined by $f(x,y)=y^{3}-x$
Then the equation $f(x,y)=y^{3}-x = 0$ generates an implicit function $g:\mathbb{R}\rightarrow \mathbb{R}$ defined by $g(x)=x^{1/3}$
However, the derivative $g'(x)=\frac{1}{3}x^{-2/3}$ is not defined at $x = 0$ because for small changes $dx$ and $dy$ we have
$df(x,y) = f_{x}(x,y)dx + f_{y}(x,y)dy = 0$
Thus
$g'(x)=dy/dx=- f_{x}(x,y)dx / f_{y}(x,y)dy$
which is defined only when
$f_{y}(x,y)dy \neq 0$
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0Could you please consider formatting your answer. Math.SE uses MathJax to display mathematical expression. I believe you have misttook the syntax. It should be in the form `$ <
>$ ` – 2012-04-11 -
0sorry, switching between SE and the browser LaTeX plugin I sometimes forget which one uses which syntax :D – 2012-04-11
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0I believe Srijan is looking for a function defined only implicitly that is non-differentiable at every point in an open interval, not just at an isolated point. See http://en.wikipedia.org/wiki/Weierstrass_function – 2012-04-11
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0right ... how about the Koch snowflake or space-filling curve – 2012-04-11
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0@BarrySmith Your particular link seems to answer my question. Are there other such examples? – 2012-04-11
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0@Srijan there are quite a few, eg: Bolzano, Cellerier, Riemann, Darboux, Peano (space-filling), Takagi, van der Waerden functions, Koch curve (snowflake) and many more http://epubl.luth.se/1402-1617/2003/320/LTU-EX-03320-SE.pdf – 2012-04-11
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0@scibuff That is indeed a great resource. – 2012-04-11