I don't know whether my doubt warrants a separate question thread as it shares similarities with my last question. So in that case I apologise.
The thing is as Siminore said a Fréchet derivative in general for infinite dimensional spaces is defined as a "Continuous linear map". Note that in this case derivative of a linear map is the map itself. So if one was asked whether its possible for a linear map to be Fréchet differentiable without being continuous, answer would be no according to me. However I came upon this on the web:
I wasn't able to access the full text and anyway very little of the title made sense to me, but the heading seemed to indicate that I was wrong, for it speaks of an everywhere discontinuous function with Fréchet derivatives. Is this possible even if my function was linear? Also how would my question make sense in case of a Gâteaux derivative which doesn't assume linearity in the first place and lastly what is the necessity for defining a Gâteaux derivative?