Page 69 - Vector Analysis

§2.8 Directional Derivatives and Gradient Vectors 65

while if v1 = 0,

(Dvf )(0) = lim f (tv1, tv2) ´ f (0, 0) = 0.
t
tÑ0

However, f is not continuous at (0, 0) since if (x, y) approaches (0, 0) along the curve x = my2

with m ‰ 0, we have

lim f (my2, y) = lim my4 m
=
yÑ0 yÑ0 m2y4 + y4 m2 + 1

which depends on m. Therefore, f is not continuous at (0, 0).

Example 2.79. Here comes another example showing that a function having directional
derivative in all directions might not be continuous. Let f : R2 Ñ R be given by

# xy if x + y2 ‰ 0 ,
f (x, y) = x + y2
0 if x + y2 = 0 ,

and v = (v1, v2) P R2 be a unit vector. Then if v1 ‰ 0,

(Dvf )(0) = lim f (tv1, tv2) ´ f (0, 0) = lim t2v1v2 = v2
t t(tv1 + t2v22)
tÑ0 tÑ0

while if v1 = 0,

(Dvf )(0) = lim f (tv1, tv2) ´ f (0, 0) = 0.
t
tÑ0

However, f is not continuous at (0, 0) since if (x, y) approaches (0, 0) along the polar curve

θ(r) = π + sin´1(r ´ mr2) 0 ă r ! 1,
2

we have

lim f (x, y) = lim r2 r2 cos θ(r) sin θ(r) = lim r(´r + mr2) sin θ(r)
sin2 θ(r) + r cos θ(r) r sin2 θ(r) ´ r + mr2
(x,y)Ñ(0,0) rÑ0+ rÑ0+

x=r cos θ(r),y=r sin θ(r)

= lim (´r + mr2) sin θ(r) = ´1
sin2 θ(r) ´ 1 + mr m
rÑ0+

which depends on m. Therefore, f is not continuous at (0, 0).

Definition 2.80. Let U Ď Rn be an open set. The derivative of a scalar function f : U Ñ R
is called the gradient of f and is denoted by gradf or ∇f .