differentiability of functions of Several Variables

differentiability of functions of Several Variables

1. From Continuity to Differentiability — The Hierarchy

For $f(x,y)$ at $(a,b)$:

but converses fail. $f$ can be continuous without partials; partials can exist without continuity; and even both partials existing does not guarantee differentiability — you need a stronger, linear approximation condition.

2. Definition — Differentiable at $(a,b)$

$f(x,y)$ is differentiable at $(a,b)$ if there are numbers $A = f_x(a,b), B = f_y(a,b)$ such that:

i.e. the error after the plane approximation goes to zero faster than the distance. Equivalently:

Geometrically: surface has a non-vertical tangent plane at $P$:

and $f$ hugs that plane to first order.

3. Sufficient Condition — The One You Will Use

If $f_x$ and $f_y$ exist and are continuous in a neighbourhood of $(a,b)$, then $f$ is differentiable at $(a,b)$ (and hence continuous). This is the practical test — check continuity of partials, not theε-δ definition.

Continuity of f_x, f_y near (a,b)  ⇒  differentiability at (a,b)  ⇒  continuity

4. Differentials and Total Derivative

• Total differential:

is the linear part of the increment $Δf$.

• If $x=x(t), y=y(t)$, then:

• If $x=x(u,v), y=y(u,v)$, then $f_u = f_x x_u + f_y y_u$ (chain rule follows from differentiability).

5. Counter-Example — Why Partials Alone Are Not Enough

f(x,y)= x y/(x²+y²) for (x,y)≠(0,0), f(0,0)=0:
 f_x(0,0)= lim h→0 [f(h,0)-0]/h =0 ; similarly f_y(0,0)=0 (both exist)
 But f is not continuous at (0,0) (limit does not exist) ⇒ cannot be differentiable.

Stronger trap:

is continuous but not differentiable at 0 (no plane).

Another classic:

has partials nowhere? At (0,0) partials do not exist.

6. Worked Examples

(a) f(x,y)= x^2 y + y sin x : f_x=2x y + y cos x, f_y= x^2 + sin x ; both continuous everywhere
    ⇒ differentiable everywhere; at (1,0): dz = (0)dx + (1)dy ; tangent plane z = (x-1)·0 + y·1 ✓

(b) f(x,y)= x^3+y^3 at (0,0): f_x=3x^2, f_y=3y^2 continuous ⇒ differentiable; plane z=0 ✓

(c) Check differentiability at (0,0) for f= x^2 y/(x^2+y^2) , f(0,0)=0:
    f_x(0,0)=0, f_y(0,0)=0 ; check limit [f(h,k)-0]/√(h²+k²)= h^2 k/(h^2+k^2)^{3/2}
    |h^2 k| ≤ (h^2+k^2)^{3/2}/(3√3)? Actually bounded by r|cos^2 sin|→0 ⇒ differentiable ✓

(d) f(x,y)= (x y^2)/(x^2+y^4), f(0,0)=0 : f_x(0,0)=0, f_y(0,0)=0 but [f(h,k)-0 -0]/r does not →0 along k^2=h ⇒ not differentiable ✓

7. Exam Points

• State definition of differentiability and tangent plane — 3-5 marks, frequent.
• Show $f_x, f_y$ continuous ⇒ differentiable ⇒ continuous — 5 marks, frequent.
• Prove $f$ not differentiable by showing discontinuity or limit in definition fails — 5-7 marks, very frequent.
• Find $dz$ and $df/dt$ via chain rule — 5 marks, frequent.
• Give example where partials exist but $f$ not differentiable / not continuous — 5 marks, almost certain.