Answer
The graph of $f(x)$ does not consist of a vertical tangent at the origin.
Work Step by Step
1. The graph of $f(x)$ has a vertical tangent at $x$ when $f(x)$ is continuous at $x$
2) The graph of $f(x)$ has a vertical tangent at $x$ when $f'(x)=\lim\limits_{h\to0}\dfrac{f(x+h)-f(x)}{h}=\pm\infty$
and $\lim\limits_{x\to0^+}f(x)=\lim\limits_{x\to0^+}(1)=1$;
Also, $\lim\limits_{x\to0^-}f(x)=\lim\limits_{x\to0^-}(-1)=-1$
As we can see that $\lim\limits_{x\to0^+}f(x)\ne\lim\limits_{x\to0^-}f(x)$, thus $\lim\limits_{x\to0}f(x)$ does not exist.
This implies that $f(x) $ is not continuous at the origin and so, the graph of $f(x)$ does not consist of a vertical tangent at the origin..