What is the precise definition about $k$ times continously differentiable at a point $\mathbf{c}\in\mathbb {R}^{m}?$ 
$k$ times continously differentiable A funcation $f$,with domain $S$ in $\mathbb {R}^{m}$ is $k$
times continously differentiable at an interior point $\mathbf {c}$ of $S$ if it and all of its first-through $(k-1)$th-order partial derivatives are continuously differentiable at $\mathbf {c}$ or,${\color{red}{\text{equivalently}}}$,if all of the first-through $k$th-order partial derivatives of $f$ exist and are continuous at every point in some
neighborhood of $\mathbf{c}$--a vector or matrix of functions is $k$
times continuously differentiable at $\mathbf{c}$ if all of its elements are $k$ times continuously differentiable at $\mathbf{c}.$

I'm really puzzled about the above definition.Conventionally,we say a funcation $f:S\rightarrow \mathbb {R}$,with domain $S$ in $\mathbb {R}^{m}$ is continuously differentiable at the point $\mathbf {c}\in \text {int} S$  if $f$ is differentiable in a neighborhood of $\mathbf {c}$,and the partial derivatives of the $f$ are continuous at $\mathbf {c}$. From this statement and the above definition " if it and all of its first-through $(k-1)$th-order partial derivatives are continuously differentiable at $\mathbf {c}$ ", I only conclude that all of the first-through $(k-1)$th-order partial derivatives of  $f$ are continuous on a neighborhood of $\mathbf{c}$, all $k$th-order partial derivatives of $f$  exist on this neighborhood and are continuous at $\mathbf{c}$.Why the above defintion said "equivalently,if all of the first-through $k$th-order partial derivatives of $f$ exist and are continuous at every point in some neighborhood of $\mathbf{c}$"? Why they are equivalent ?  Further ,what is the precise definition about $k$ times continously differentiable at a point $\mathbf{c}\in\mathbb {R}^{m}?$
 A: tl; dr: A small sample of authors do not use the phrase continuously differentiable at a point, but only speak of continuous differentiability on an open set. The stated definition is consistent with this usage.

When $k = 1$, the definition reads "A function $f$ [...] is continously differentiable at an interior point $c$ if it is continuously differentiable at $c$ or, equivalently, if all of the first-order partial derivatives of $f$ exist and are continuous at every point in some neighborhood of $c$."
The first part is tautological, but the second serves as a base for a recursive definition.
When $k = 2$, the definition reads "[$f$] is [twice-]continously differentiable at an interior point $c$ if it and all its first partial derivatives are continuously differentiable at $c$ or, equivalently, if all of the first- and second-order partial derivatives of $f$ exist and are continuous at every point in some neighborhood of $c$."
This should make clear that

*

*According to this definition, continuous differentiability and its higher-order analogues are local conditions, not pointwise conditions.

*This definition is internally consistent. The contrary appearance comes from using the phrase "at $c$" to connote a condition holding near $c$.


To me, continuous differentiability is a global condition in the same way continuity is: It asserts a property (existence and continuity of the partial derivatives, which implies existence and continuity of the derivative) at each interior point of the domain.
I haven't made a careful study of how "$k$-times continuously differentiable" is defined by various authors, but two nearby books (Vector Calculus, Fourth Edition by Susan Colley, Definition 4.4, p. 138; and Multivariable Mathematics by MSE's own Ted Shifrin, p. 120) concur that "a function $f$ is $k$-times continuously differentiable on an open set $U$ if all the partial derivatives up to and including order $k$ exist and are continuous on $U$".
