 Higher-order differentiability at a point

For $n=1$, differentiability at $a$ means that the difference quotient has a finite two-sided limit of a function. For $n\ge 2$, require derivatives through order $n-1$ throughout some open neighbourhood of $a$, and require the final difference quotientto exist as a finite real number. The last derivative need not exist elsewhere or be continuous at $a$. In particular, existence of $f^{\prime }(a)$ by itself does not justify taking a second derivative: the function $f^{\prime }$ must first be defined in a neighbourhood of $a$.