1

For a function of one variable just looking at the graph I can see that the condition for a one-to-one analytic function is:

$$y=f(x)$$

is one-to-one if we have the condition:

$f'(x)>0$ for all $x$. i.e. the function is always increasing.

Then one might write $f'(x) = g(x)^2$ for some function $g$. Then we could write that a general function $f$ is a one-to-one:

$$f(x)\equiv \int^x_0 g(x)^2 dx$$

How would we generalise this to multiple variables? e.g. $y_n = f_n(x_1,..x_n)$

zooby
  • 4,343
  • 1
    For multiple variables we have the Jacobian, the matrix of partial derivatives. If the determinant is nonzero, we have a local diffeomorphism, by the inverse function theorem. –  Feb 21 '20 at 08:03
  • @Chris thanks!! – zooby Feb 21 '20 at 18:01

0 Answers0