In the context of maps between Banach spaces, a map $f:E\to F$ is differentiable at $x\in E $ if it is continuous at $ x $ and there exists a linear map $ T:E\to F $ such that $$\lim_{h\to0}\frac{\lVert f(x+h)-f(x)-Th\rVert}{\lVert h\rVert}=0.$$
The continuity assumption here then implies that the linear map $ T $ is bounded. Also, if one assumes in the definition that $ T $ is a bounded linear map, then $ f $ will be continuous at $x$.
I was wondering if someone had in mind an example of Banach spaces $ E, F $, a map $f:E\to F $, which isn't continuous at $ x\in E $, and an unbounded linear map $ T:E\to F $ such that $$\lim_{h\to0}\frac{\lVert f(x+h)-f(x)-Th\rVert}{\lVert h\rVert}=0.$$
Such an example would motivate the necessity of including either $ f$ continuous at $ x $ or $ T $ bounded in the definition of differentiability.
Best Answer
Let $f \colon E \to F$ be an unbounded linear map. Then $f$ is not continuous, but differentiable with $Df(x) = f$ for each $x \in E$.