Let me state my problem. Suppose we have a ball $B$ in standard $\mathbb{R}^3$, that is a $\varepsilon$-neighbourhood of $0$ point. Suppose we have a family of cones $X_C = \lbrace C > 0 \vert x^2 + y^2 \leqslant C \cdot z^2 \rbrace $. Also we have a homeomorphism $h$ that maps $B$ on itself ($h(B) = B$) and $h(0) = 0$ (if it's crucial, $h(B \cap Oz) = B \cap Oz$ and $h(0) = 0$).
So, the question is: if we take a cone, corresponding to some value $C_1$, does exist some $C_2$ that $f(X_{C_1}) \subseteq X_{C_2}$?