Suppose we have two functions: $f_1(x)=x^2,x\geq 0$ and $f_1(x)=0,x\leq 0$ and $f_2(x)=0,x\geq 0$ and $f_2(x)=x^2, x\leq0$. Show that these two functions $f_1(x)$ and $f_2(x)$ are linearly independent.
My attempt: Consider the interval $[0,\infty]$. Then $f_1(x)=x^2$ and $f_2(x)=0$. The Wronskian $W(f_1,f_2)=0$. Since the Wronskian is zero for every $x\in [0,\infty]$, we conclude that over this interval the functions are linearly dependent. Similar reasoning shows that for $x\in[-\infty,0]$ the functions are linearly dependent. Thus, one might conclude for all $x$ the functions are linearly dependent. However, this is not true. What is wrong with my argument?
I know that the Wronskian being equal to zero is not a sufficient condition of linear dependency, but then several solutions make use of this condition directly to establish linear independency.