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Prove the zero derivative theorem for vector valued functions

by RoRi

Prove the zero derivative theorem for vector valued functions. In other words, show that if F'(t) = 0 for each t in an open interval I, then there exists a vector C such that F(t) = C for all t \in I.

Proof. If F'(t) = 0 then we have

    \begin{align*} && (f_1'(t), \ldots, f_n'(t)) &= (0, \ldots, 0) \\ \implies && f_i'(t) &= 0 & \text{for all } i. \end{align*}

By the zero derivative for real valued functions we then have f_i (t) = c_i for some constant c_i for each i = 1, \ldots, n. Therefore we have

    \[ (f_1(t), \ldots, f_n (t)) = (c_1, \ldots, c_n) \quad \implies \quad F(t) = C. \qquad \blacksquare \]

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