3.3. DERIVATIVES237A third example is defined by a straightforward formula (no “cases”):f(x,y) =x1/3y1/3.Again, the function is constant along the coordinate axes, so both partialsare zero. However, if we try to evaluate the limit in Equation (3.6) usingany vector not pointing along the axes, we getd(0,0)f(α−→ı+β−→) =ddtvextendsinglevextendsinglevextendsinglevextendsinglet=0bracketleftBigα1/3β1/3t2/3bracketrightBig;sincet2/3is definitely not differentiable att= 0, the required linear mapd(0,0)f(α−→ı+β−→) cannot exist.From all of this, we see that having the partials at a point−→x0is notenough to guarantee differentiability off(−→x) at−→x=−→x0. It is not evenenough to also have partials at every point near−→x0—all our examplesabove have this property. However, a slight tweaking of this last conditiondoesguarantee differentiability. We callf(−→x)continuouslydifferentiableat−→x0
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