 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/ib/paper-3/3b/solution

a_{0} = (a, b)

if there is a linear map

L : R^{2} \to R

with

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

L = D f_{a_{0}}

is its derivative; equivalently

L (h_{1}, h_{2}) = A h_{1} + B h_{2}

for fixed constants. Since every such linear map is bounded,

∣ f (a_{0} + h) - f (a_{0}) ∣ \leq ∥ L ∥∥ h ∥ + ∣ r (h) ∣ \to 0

. This proves differentiability implies continuity, not merely continuity along each coordinate axis.

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