 Solution

ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2005/f/1/i/solution

D f (a) : R^{n} \to R^{n}

such that

f (a + h) = f (a) + D f (a) h + r (h), \frac{∥ r ( h ) ∥}{∥ h ∥} ⟶ 0 (h \to 0) .

It is unique: the difference of two such linear maps, evaluated on

h = t v

, must vanish on every fixed vector

v

after division by

t

and passage to the limit.

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