Showing that $\{(x,|x|):x\in\mathbb{R}\}$ is not the image of an immersion

Question

Show that $A=\{(x,|x|):x\in\mathbb{R}\}$ is not the image of an immersion $f:\mathbb{R}\to\mathbb{R}^2$.

Here are the definitions I know:

• Let $M$ be a differentiable manifold. A tangent vector at $p\in M$ is a linear function $v:C^{\infty}(M)\to\mathbb{R}$ that is Leibnizian at $p$. The collection of these is the tanget space $T_pM$.
• The differential map of a differentiable map $f:M\to N$ at a point $p\in M$ is the function $df_p:T_pM\to T_{f(p)}N$ given by $df_p(v)(g)=v(g\circ f)$ where $v\in T_pM$ and $g\in C^{\infty}(N)$.
• An immersion is a differentiable map $f:M\to N$ such that $df_p$ is injective for every $p\in M$.

Thus, if we suppose that $f:\mathbb{R}\to\mathbb{R}^2$ is a differentiable map with $f(\mathbb{R})=A$, we should show that $df_p$ is not injective for some $p\in\mathbb{R}$. I really don't know what to do, but maybe we should use that $x\to |x|$ is not differentiable in the usual sense at $x=0$, but what tangent vectors should we take in order to show $df_p$ is not injective?

Thank you.