Equivalent conditions for a linear map between two normed spaces to be continuous everywhere/2 implies 3

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See page 154 in Maurin's Analysis, although the proof isn't hard

Statement

Given two normed spaces $(X,\Vert\cdot\Vert_X)$ and $(Y,\Vert\cdot\Vert_Y)$ and also a linear map $L:X\rightarrow Y$ then we have:

• If {{M|L}] is continuous at a point (say $p\in X$) then
• $L$ is a bounded linear map, that is to say:
  • $\exists A\ge 0\ \forall x\in X[\Vert L(x)\Vert_Y \le A\Vert x\Vert_X]$