Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
add comment

1 Answer 1

up vote 0 down vote accepted

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

$f_{y}(x,y)dy \neq 0$

share|improve this answer
    
Could 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 $ <<Tex Commnad here>>$ –  Bidit Acharya Apr 11 '12 at 13:34
    
sorry, switching between SE and the browser LaTeX plugin I sometimes forget which one uses which syntax :D –  scibuff Apr 11 '12 at 13:39
    
I 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 en.wikipedia.org/wiki/Weierstrass_function –  Barry Smith Apr 11 '12 at 13:48
    
right ... how about the Koch snowflake or space-filling curve –  scibuff Apr 11 '12 at 14:16
    
@BarrySmith Your particular link seems to answer my question. Are there other such examples? –  Srijan Apr 11 '12 at 16:16
show 2 more comments

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.