- Calculus, Volume 1: One Variable Calculus, with an Introduction to Linear Algebra
- Tom M. Apostol
- Second Edition
- 1967
- 978-1-119-49673-1
14.4 Exercises
- Prove that a vector-valued function $F$ is differentiable on an open interval $I$ if and only if for each $t$ in $I$ we have:$$F'(t) = \lim_{h\to 0} \frac{1}{h}[F(t + h) - F(t)]$$
- By definition, a vector-valued function is differentiable on an open interval $I$ if and only if every component is differentiable on that open interval. In other words, for each component $f$ of $F,$ $$f'(t) = \lim_{h \to 0}\frac{1}{h}\left[f(t + h) - f(t)\right]$$for all $t$ in the open interval $I.\quad\blacksquare$