Definition (in our lecture): Let $X,Y$ be normed vector spaces and $f:U \to Y$ with $U \subset X$ open. Then $f$ is called Fréchet-differentiable in $x_0 \in U$ if there exists a linear map $A:X \to Y$ such that
$$\lim_{x \to x_0} \frac{||f(x)-f(x_0)-A(x-x_0)||_Y}{||x-x_0||_X}=0.$$
In all books I looked into there is additionally required that $A$ is continuous. Is this assumption necessary?