3.3. DERIVATIVES
237
A third example is defined by a straightforward formula (no “cases”):
f
(
x,y
) =
x
1
/
3
y
1
/
3
.
Again, the function is constant along the coordinate axes, so both partials
are zero. However, if we try to evaluate the limit in Equation (
3.6
) using
any vector not pointing along the axes, we get
d
(0
,
0)
f
(
α
−→
ı
+
β
−→
) =
d
dt
vextendsingle
vextendsingle
vextendsingle
vextendsingle
t
=0
bracketleftBig
α
1
/
3
β
1
/
3
t
2
/
3
bracketrightBig
;
since
t
2
/
3
is definitely not differentiable at
t
= 0, the required linear map
d
(0
,
0)
f
(
α
−→
ı
+
β
−→
) cannot exist.
From all of this, we see that having the partials at a point
−→
x
0
is not
enough to guarantee differentiability of
f
(
−→
x
) at
−→
x
=
−→
x
0
. It is not even
enough to also have partials at every point near
−→
x
0
—all our examples
above have this property. However, a slight tweaking of this last condition
does
guarantee differentiability. We call
f
(
−→
x
)
continuously
differentiable
at
−→
x
0
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 Fall '08
 ALL
 Calculus, Derivative, required linear map, △x

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