Determination of Invertibility

Question

If for example $f(x_1,x_2,x_3,x_4)= x_1- \sin(x_3)$ is a nonlinear function of $(x_1,x_2,x_3,x_4)$, how can be understood that for example $\frac{df(x_1,x_2,x_3,x_4)}{dx_3}$ is invertible or not ?

Answer 1

There is no straightforward, always-working, follow-these-steps-to-get-what-you-need answer to your question. In general, proving that a function is invertible can be a very hard thing to do.

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However, in your case, it's simple. Since $f$ is independent of $x_2$, it is not invertible (for example, $f(0,0,0,0) = f(0,1,0,0)$). The same is true for all its derivatives.