# Definition If a [[map|function]] on a [[manifold]] $M$, $f:M\rightarrow \mathbb{R}^n$ is *differentiable* on $\mathbb{R}^n$ and maps regions of $M$ differentiably to $\mathbb{R}^n$, then it is said to be a differentiable function on $\mathbb{R}^n$ ([[Real Coordinate Space|R^n]]) More abstractly, the function may be written as $f(P), \, P \in M$. But $P$ has [[Manifold#^e43e20|coordinates]], so one can express the value of the function by some algebraic expression $f(x^i)$. If this expression is differentiable in all its arguments, then the function is *differentiable*. The coordinates themselves, are, of course, [[Continuous Map|continuous]] and infinitely [[Differentiable Function|differentiable functions]], assuming a $C^\infty$-manifold. # In Physics In physics, one often assumes that we can place coordinates $\{x^i, i = 1\ldots n\}$ on the [[manifold]], and that any sufficiently-differentiable set of equations $y^i = y^i(x^j)$, which is locally-invertible (i.e. the [[Jacobian]] $|\partial y^i/\partial x_j| \neq 0$) constitutes an acceptable [[Coordinate Transformations on Manifolds|coordinate transformation]] to new coordinates $\{y^i, i = 1\ldots n\}$. We often avoid referring to the mapping directly.