Inverse function in one dimension

[zetafunction]

can a function in ONE dimension have NO inverse ?? i mean

if given the inverse function $$f^{-1} (x) = g(x) + \sum_{k=-N}^{k=N}c_{k}exp(ixlogk)$$

the first function g(x) is an smooth function , the last Fourier series is a 'noise correction' t o this function g , N is a big but finite number (otherwise the OFurier series could diverge)

how could i use numerical methods to get f(x) ??

g(x) is always an INCREASING function.

 

[g_edgar]

If a function is not one-to-one, then it has no inverse. Whether yours is one-to-one probably depends on the coefficients.

 

[zetafunction]

however the sine function and x^2 are not strictly one to one but one can define an inverse

 

[Mute]

however the sine function and x^2 are not strictly one to one but one can define an inverse

But we can only define the inverses on domains for which the functions are 1 to 1.

 

[lavinia]

can a function in ONE dimension have NO inverse ?? i mean

if given the inverse function $$f^{-1} (x) = g(x) + \sum_{k=-N}^{k=N}c_{k}exp(ixlogk)$$

the first function g(x) is an smooth function , the last Fourier series is a 'noise correction' t o this function g , N is a big but finite number (otherwise the OFurier series could diverge)

how could i use numerical methods to get f(x) ??

g(x) is always an INCREASING function.

there are continuous functions that are not one to one on any interval. Such functions can not be differentiable at any point.