$C^k$ Euclidean maps are closed under componentwise algebra and composition

Given: Open Euclidean domains and maps for which the displayed sums, products, scalar multiples, or composites are defined. We use induction on derivative order (The principle of mathematical induction).

A map $f:U\to {\mathbb{R}}^q$ is of class $C^k$ when each component is of class $C^k$ ($C^k$ Euclidean maps and diffeomorphisms).

If $f$ is totally differentiable at $a$ and $g$ is totally differentiable at $f(a)$, then $g\circ f$ is totally differentiable at $a$ and $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$ (The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$).

For differentiable real functions, sums and scalar multiples are differentiable with the expected derivatives, and $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$ (Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$).

If every first partial derivative exists near a point and is continuous there, then the map is totally differentiable there and its total derivative has matrix equal to its Jacobian (If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative).

Finite sums, products, and scalar multiples of continuous real functions are continuous (Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function), and a composite of continuous maps between topological spaces is continuous (Continuity may be checked on any open cover, and on any finite closed cover; composites of continuous maps are continuous, claim 1).

Total differentiability gives every directional and partial derivative by applying the total derivative to the corresponding direction vector (A total derivative computes every directional derivative, and its matrix is the Jacobian).