Need to show that an injective map between manifolds need not be an immersion. $x^3$ example was given here but wanted to check if my justification suffices. https://mathoverflow.net/questions/15317/is-an-injective-smooth-map-an-immersion
$f:R\rightarrow R$ ; $f(x)=x^3$. $f$ is an immersion iff $\forall x\in R$, $df_x$ is an injective linear transformation and for $x^3$, $df_0$ is a $zero$ transformation and hence not an injective linear transformation. Thus not an immersion? Thanks!