It says for V and V' under two different normed spaces, let T:V->V' be a linear map. Then T is continuous at 0 implies T is continuous. But then they say
''Since $\left \| Tx-Ty \right \|' = \left \| T(x-y) \right \|'<\epsilon$
 , T is continuous at all y in V ''. 

So I dont understand where they have used the fact that T is continuous at 0 as if it were wouldnt we use $\left \| Tx \right \|'<\epsilon$ only to deduce that its continuous at any y.