embedding continuous function as diffeomorphism

Question

For any $\varepsilon>0$, any continues function $f$ from $[0,1]^2\times \{0\}$ to $\mathbb{R}^2 \times \{0\}$, can we find a diffeomophism $\Phi$ of $\mathbb{R}^3$ such that $$ \|f(x) - \Phi(x)\| < \varepsilon, \forall x \in [0,1]^2\times \{0\}? $$ Note: $f$ comes from a function $\tilde f: [0,1]^2 \to \mathbb{R}^2$ by filling a zero in the input and output coordinates. According to the Stone-Weierstrass theorem, we only need to consider $f$ as a polynomial.