# Differentiability implies continuity in Banach space

I am reading a book on differential calculus in normed linear space and something escapes me about how to define differentiability.

I consider $$f : U\to E$$ where U is an open subset of $$V$$ and $$V$$ and $$E$$ are two Banach spaces.

We say that $$f$$ is differentiable on $$U$$ if there exists $$\Lambda\in L(V,E)$$ such that $$f(x+h) - f(x) - \Lambda(h) = o(h)$$ . We want to show that this implies continuity

Consider $$\epsilon>0$$, take $$0<\delta<1$$ such that if $$\lVert h\rVert_{V}\leq\delta$$ then $$x+h\in U$$ and $$\lVert f(x+h) - f(x) - \Lambda(h)\rVert_{E}\leq \lVert h\rVert_{V}\quad (1)$$

This requirement seems clear for me. In fact it relies on the notion of tangent in Banach space and it is explained in the text as follows : $$f$$ and $$g$$ are said to be tangent to one another at $$x_0$$ if $$\forall\epsilon>0, \exists r>0 : \lVert h\rVert < r \implies \lVert f(x_0+h) - g(x_0+h)\rVert\leq\epsilon\lVert h\rVert$$

And in the definition of the differentiability introduce in the text this "tangent property" is satisfied so this inequality makes sense.

Now, here is the point I don't understand after $$(1)$$ : Consequently, $$\lVert f(x+h) - f(x)\rVert_{E}\leq (\lVert\Lambda\rVert_{E} +1)\lVert h\rVert_{V}$$

And I don't see where does it come from, someone has an idea ? Thank you a lot !

Using $$\lVert f(x+h) - f(x) - \Lambda(h)\rVert_{E}\leq \lVert h\rVert_{V}\quad (1)$$ and triangle inequality we get $$\lVert f(x+h) - f(x) \leq \lVert h\rVert_{V} +\lVert\Lambda(h)\rVert_{E}$$ and $$\|\Lambda (h)\|_E\leq \|\Lambda \|_E \|h\|_V$$.
• Thank you a lot for your answer. We have $0\leq\lVert f(x+h) - f(x) - \Lambda(h)\rVert_{E}\leq\lVert h\rVert_{V}$ So how can we insure that by the triangle inequality we don't have this issue $\lVert f(x+h) - f(x)\rVert_{E} + \lVert\Lambda(h)\rVert_{E}\geq \lVert h\rVert_{V}$ ? And also I don't see how you get $\lVert\Lambda(h)\rVert_{E}\leq\lVert\Lambda\rVert_{E}\lVert h\rVert_{V}$ ? Thank you Dec 22, 2022 at 23:36
• For the first question write $f(x+h) - f(x)$ as $[f(x+h) - f(x) - \Lambda(h)]+\Lambda(h)$ and apply triangle inequlaity. For the second question $\|Tx\|\leq ||T\|\|x\|$ for any bounded operator on a normed linear space. This is an easy consequence of the definition of norm of an operator. @coboy Dec 22, 2022 at 23:47