Gateux differential question

Question

I am going over the book "Optimization by Vector Space Methods" by Luenberger, and I found a statement that I'm not sure I understand. I am attaching the relevant page and I drew a box around the question in hand. I am not sure why the Gateux differential equals $\frac{d f(x+\alpha h)}{d \alpha}$ at $\alpha=0$ in the case the functional is real.

Thanks in advance

Answer 1

By definition, $$ \delta f(x,h)=\lim\limits_{\alpha\to 0}\frac {f(x+\alpha h)-f(x)}{\alpha}=\lim\limits_{\alpha\to 0}\frac {f(x+\alpha h)-f(x+0h)}{\alpha} $$ which is by definition the derivative of the function $\alpha\to f(x+\alpha h)$ at $\alpha=0$.