720 CHAPTER 22. THE DERIVATIVE

Theorem 22.6.6 Suppose f : U → Y where U is an open set in X , a normed linear space.Suppose that f is Gateaux differentiable on U and that the Gateaux derivative is continuousat x. Then f is Frechet differentiable at x and Df(x)v = Dvf(x).

Example 22.6.7 Let X be C20(Ω̄)

where Ω is a bounded open set in Rnconsisting of thosefunctions which are twice continuously differentiable and vanish near ∂Ω. The norm willbe

∥u∥X ≡ ∥u∥∞+max{∥u,i ∥∞

, i}+max{∥∥u,i j

∥∥∞

i, j}

Then let f : X → R be defined by

f (u)≡ 12

∫Ω

∇u ·∇udx

Show f is differentiable at u ∈ X.

Consider the Gateaux differentiability.

limt→0

f (u+ tv)− f (u)t

= limt→0

t∫

Ω∇u ·∇vdx

t+ t

12

∫Ω

∇v ·∇v

so it converges to ∫Ω

∇u ·∇vdx =−∫

Ω

∆uvdx

the last step comes from the divergence theorem. Clearly v→−∫

Ω∆uvdx is linear and R

valued. ∣∣∣∣−∫Ω

∆uvdx∣∣∣∣≤ ∥v∥X

∫Ω

|∆u|dx≤ ∥v∥X m(Ω)∥u∥X

Thus this appears to be in L (X ,R). This also shows that,

sup∥v∥≤1

|Dv f (u)−Dv f (û)| ≤ m(Ω)∥u− û∥X

and so u→ D(·) (u) is continuous as a map from X to L (X ,R) so it seems that this is adifferentiable function and

D f (u)(v) =−∫

Ω

∆uvdx

Definition 22.6.8 Let f : U → Y where U is an open set in X. Then f is called C1 (U) if itGateaux differentiable and the Gateaux derivative is continuous on U.

As shown, this implies f is differentiable and the Gateaux derivative is the Frechetderivative. It is good to keep in mind the following simple example or variations of it.

Example 22.6.9 Define

f (x)≡{

x2 sin( 1

x

)x ̸= 0

0 if x = 0

This function has the property that it is differentiable everywhere but is not C1 (R). Infact the derivative fails to be continuous at 0.