real analysis - Number of Functions satisfying linear homogeneous function criterion

Let us suppose I have a continuous smooth function

$f : \mathbb R^n\rightarrow \mathbb R$

Satisfying

$f(\lambda x_1,\lambda x_2, \ldots ,\lambda x_n)= \lambda f( x_1,x_2, \ldots , x_n),\space \forall\lambda\in\mathbb R \space \ldots(linear \space condition)$

My objective is to find the function $f$

Attempt:

Expand in Taylor series the function $f$ for $n$ variables $(i.e.\space x_i)\space in \space the\space equation\space of \space linear \space condition$

1)The constant term is zero as $f(0,...,0)=0$.

2)The linear terms cancel either side.

3)With the higher order terms of coefficients c, $$\lambda^{k-1}c_k=c_k$$

This condition holds true $\forall k,\lambda \in \mathbb R$

This implies that all coefficients of higher order(>1) terms are zero.

Hence I conclude that the function is $$f( x_1,x_2, \ldots , x_n)= a_1 x_1 + a_2 x_2 \ldots +a_n x_n$$

for some constants $a_i(1\leq i \leq n)$

My doubt is:

1) Is my reasoning correct, or is there a better way to do it?

2) Is this the only unique solution for the function?