Smooth functions

Recall that if $f \colon U \to \mathbb{R}$ is a function defined on an open subset $U$ of $\mathbb{R}$ then we say that $f$ is differentiable at $x \in U$ if the limit

$$\lim_{h \to 0} \frac{ f(x+h) - f(x) }{ h }$$

exists. If the limit exists we call it the derivative of $f$ at $x$ and denote it by any of

$$df(x), \quad d_xf,$$   or$$\quad f'(x).$$

If $f$ is differentiable at any $x$ in $U$ we just say that $f$ is differentiable.

If $U \subset \mathbb{R}^n$ is open and $f \colon U \to \mathbb{R}$ we can define partial derivatives by varying only one of the co-ordinates. If $e^i$ is the element of $\mathbb{R}^n$ with a $1$ in the $i$th position and 0's elsewhere we define a curve by

$$\gamma_i(t) = x + te_i.$$

The $i$th partial derivative of $f$ at $x$ is then defined by

$${\partial _i f }(x) = (f\circ\gamma_i)'(0).$$

We say that $f$ is smooth if it has partial derivative of any order. Because a differentiable function is continuous it follows that $f$ has continuous partial derivative of any orders. I am quite deliberately avoiding the notation

$$\frac{\partial f}{\partial x^i}(x)$$

for the time being.

If $f\colon U \to \mathbb{R}^m$ then we say that $f$ is smooth if the functions $f^i \colon U \to \mathbb{R}$ are smooth where $f(x) = (f^1(x), f^2(x), \dots, f^m(x))$. Notice that in this case the limit definition of derivative makes sense and we can define

$$f'(x) = df(x) = d_xf = \lim_{h \to 0} \frac{ f(x+h) - f(x) }{ h }.$$