On some attributes of arbitrary functions involved in the general solution provided by Lagrange's method of characteristics

Question

I met with a M.S.Q question [CSIR India June 2011];

Q. A general solution of the PDE $u u_x+y u_y=x$ is of the form,

1. $f\left(u^2-x^2, \frac{y}{x+u}\right)=0$, where $f: \mathbf{R}^2 \rightarrow \mathbf{R}$ is $C^1$ and $\nabla f \neq(0,0)$ at every point

2. $u^2=g\left(\frac{y}{x+u}\right)+x^2, g \in C^1(\mathbf{R})$

3. $f\left(u^2+x^2\right)=0, f \in C^1(\mathbf{R})$

4. $f(x+y)=0, f \in C^1(\mathbf{R})$

I used the Lagrange's method of characteristics to get the general solution in the form $$f(u^2-x^2,\frac{y}{x+u})=0,$$ where $f\in C^1(\mathbf R^2)$,

or, $$u^2=x^2+g(\frac{y}{x+u}),$$where $g \in C^1(\mathbf R)$. Also, I can eliminate options 3 and 4.

However, what does '$\nabla f\ne 0$ at every point' mean? What's the role in?

Nevertheless, How can we ensure the $C^1$-ness of $f$ and $g$ throughout $\mathbf R^k$?

Thanks in advance,