I have the following definition of Gateaux differentiability
$f$ is Gateaux differentiable at $x_0$ if there is a continuous and linear operator $T$ so that
$$ \lim_{t \rightarrow 0}\frac{f(x_0+th)-f(x_0)}{t}=Th \qquad\forall h \in X $$ but in my opinion this means that $f$ is continuously differentiable as $T$ is continuous, but this cannot be true.
so what is the difference between a continuous operator and continous differentiability
Saying that $f$ is continuous differentiable means something more: suppose that $f$ is Gateaux continuously differentiable in some neighbourhood $U \ni x_0$: it means that for each $y \in U$ there's a continuous linear operator $T_y$, such that $\lim_{t \to 0} \frac{f(y + th) - f(y)}{t} = T_y(h)$ for all $h \in X$, and the function $y \mapsto T_y$ is continuous. Generally speaking, if $f$ is not continuously differentiable, the mapping $y \mapsto T_y$ need not be continuous.