Theorem The Fréchet derivative is unique.
Proof. Assume that both $A$ and $B$ in satisfy the condition for the Fréchet derivative at the point $\mathbf{x}$ . To prove that they are equal we will show that for all $\varepsilon >0$ the operator norm $\|A-B\|$ is not greater than $\varepsilon$ . By the definition of limit there exists a positive $\delta$ such that for all $\|\mathbf{h}\|\leq\delta$ $$ \|f(\mathbf{x}+\mathbf{h})-f(\mathbf{x})-A\mathbf{h}\|\leq\frac{\varepsilon}{2}\cdot\|\mathbf{h}\| \mbox{ and } \|f(\mathbf{x}+\mathbf{h})-f(\mathbf{x})-B\mathbf{h}\|\leq\frac{\varepsilon}{2}\cdot\|\mathbf{h}\ $$ holds. This gives