Definition. Let $X,Y$ be Banach spaces and $f:U\rightarrow Y$ a map on an open subset $U\subset X$.
The map $f$ is called Gâteaux differential at $x\in U$ if there exists a continous linear map $A:X\rightarrow Y$ such that $$\displaystyle\lim_{t\rightarrow 0}\dfrac{f(x+td)-f(x)}{t}=Ad,\forall d\in X$$ The $A$ then is called Gâteaux derivative of $f$.
I don't get the intuition ideal behind the condition $A$ is linear (and moreover continous). Why we define it that way and what happens if $A$ is not linear (or continous)? (particularly in finite and infinite dimensional spaces)