Questions: Differentiability in Multiple Variables

Question 1

Consider f(x,y) = xy/(x²+y²) for (x,y) ≠ (0,0) and f(0,0) = 0. Both partial derivatives fₓ(0,0) and f_y(0,0) equal 0. Is f differentiable at the origin?

• A. Yes, because both partial derivatives exist at the origin
• B. Yes, because fₓ = f_y = 0 confirms a horizontal tangent plane z = 0
• C. No, because the partial derivatives don't exist at the origin
• D. No, because the limit of f(x,y) as (x,y)→(0,0) depends on the direction of approach — along y = x the function equals 1/2, not 0

Answer: (D)

The existence of partial derivatives is necessary but not sufficient for differentiability. Partial derivatives only probe the function along axis-aligned directions (the x- and y-axes). For differentiability, the tangent plane approximation must work in *all* directions simultaneously. Along y = x, f(x,x) = x²/2x² = 1/2 for all x ≠ 0, so f does not approach 0 as required for the tangent plane z = 0 to be a valid approximation. Options A and B represent the classic misconception that 'existing partial derivatives = differentiable.'

Question 2

Which chain of implications correctly describes the relationship between continuity, differentiability, and continuous partial derivatives for multivariable functions?

• A. Continuous ⟹ differentiable ⟹ continuous partial derivatives
• B. Continuous partial derivatives ⟹ differentiable ⟹ continuous
• C. Differentiable ⟹ continuous partial derivatives ⟹ continuous
• D. Continuous ⟹ continuous partial derivatives ⟹ differentiable

Answer: (B)

The correct chain is: continuous partial derivatives ⟹ differentiable ⟹ continuous. None of the arrows reverses. Continuous partials are a sufficient (not necessary) condition for differentiability. Differentiability implies continuity but not vice versa. Options A and D both incorrectly place continuity at the start — mere continuity implies nothing about derivatives in the multivariable setting.

Question 3

A function f(x,y) that is differentiable at (a,b) must be continuous at (a,b).

• T. True
• F. False

Answer: True

Differentiability implies continuity in both single-variable and multivariable calculus. If the tangent plane approximation L(x,y) is valid (error going to zero faster than distance), then as (x,y)→(a,b), f(x,y) must approach L(a,b) = f(a,b), which is exactly the definition of continuity. This implication holds; the reverse does not — a function can be continuous at a point yet not differentiable there.

Question 4

If both partial derivatives fₓ(a,b) and f_y(a,b) exist, then f is differentiable at (a,b).

• T. True
• F. False

Answer: False

This is the central misconception of multivariable differentiability. Partial derivatives only measure rates of change along the coordinate axes — they say nothing about behavior in other directions. The classic counterexample is f(x,y) = xy/(x²+y²) at the origin: both partials exist and equal 0, yet the function is not differentiable there because its behavior along diagonal directions contradicts the tangent plane. The correct sufficient condition requires that the partial derivatives *exist and are continuous* in a neighborhood of (a,b).

Question 5

Why is the existence of partial derivatives at a point insufficient to guarantee differentiability? What additional condition is sufficient?

Model answer: Partial derivatives only measure the function's rate of change along axis-aligned directions (parallel to the x- or y-axis). Differentiability requires that the tangent plane approximation be valid for *all* directions of approach — not just the coordinate directions. A function can have well-defined partial derivatives yet behave pathologically along diagonal or other directions. A sufficient condition is that the partial derivatives exist *and are continuous* in a neighborhood of the point; this guarantees that the function is approximable by its tangent plane from all directions simultaneously.

The key distinction is between 'probing along two special directions' (what partial derivatives do) and 'being approximable by a linear function in every direction' (what differentiability requires). This is precisely why directional derivatives in non-axis directions are not automatically determined by the partial derivatives unless differentiability holds — and why differentiability is the correct generalization of smoothness to multiple variables.