470 CHAPTER 22. DERIVATIVE OF A FUNCTIONS OF MANY VARIABLES

Taking the partial derivatives of f , fx = y, fy = x, fz = 2z. These are all continuous.Therefore, the function has a derivative and fx (1,2,3) = 1, fy (1,2,3) = 2, and fz (1,2,3) =6. Therefore, D f (1,2,3) is given by D f (1,2,3) = (1,2,6) . Also, for (x,y,z) close to(1,2,3),

f (x,y,z) ≈ f (1,2,3)+1(x−1)+2(y−2)+6(z−3)= 11+1(x−1)+2(y−2)+6(z−3) =−12+ x+2y+6z

When a function is differentiable at x0, it follows the function must be continuous there.This is the content of the following important lemma.

Lemma 22.4.6 Let f : U → Rq where U is an open subset of Rp. If f is differentiableat x, then f is continuous at x. In fact, there is a constant C such that if |v| is sufficientlysmall, then |f (x+v)−f (x)| ≤C |v|.

Proof: From the definition of what it means to be differentiable and Lemma 22.2.2, if|v| is small enough,

|f (x+v)−f (x)|= |Df (x)(v)+o(v)| ≤ |Df (x)(v)|+ |v| ≤ C̃ |v|+ |v| ≡C |v|

Note that this also says that if |v| is small enough, then

|f (x+v)−f (x)||v|

≤C (22.8)

There have been quite a few terms defined. First there was the concept of continuity.Next the concept of partial or directional derivative. Next there was the concept of differ-entiability and the derivative being a linear transformation determined by a certain matrix.Finally, it was shown that if a function is C1, then it has a derivative. To give a rough ideaof the relationships of these topics, here is a picture.

Continuous|x|+ |y|

Partial derivativesxy

x2+y2

derivative

C1

You might ask whether there are examples of functions which are differentiable butnot C1. Of course there are. In fact, Example 22.2.9 is just such an example as explainedearlier. Then you should verify that f ′ (x) exists for all x ∈ R but f ′ fails to be continuousat x = 0. Thus the function is differentiable at every point of R but fails to be C1 becausethe derivative is not continuous at 0.

Example 22.4.7 Find an example of a function which is not differentiable at (0,0) eventhough both partial derivatives exist at this point and the function is continuous at thispoint.

Here is a simple example.

f (x,y)≡

{xsin

(1xy

)if xy ̸= 0

0 if xy = 0