Questions: Differentiability in Multivariable Functions

Question 1

Function f has both partial derivatives fₓ(0,0) = 0 and f_y(0,0) = 0, yet f is not continuous at (0,0). What does this imply?

• A. This is impossible — if both partial derivatives exist, continuity is guaranteed
• B. f is differentiable at (0,0) because both partial derivatives exist
• C. f is not differentiable at (0,0), since differentiability implies continuity and continuity fails
• D. f is differentiable but the tangent plane formula does not apply

Answer: (C)

Differentiability is a strictly stronger condition than the existence of partial derivatives. The hierarchy is: differentiable ⟹ continuous ⟹ (partial derivatives may exist). Since differentiability implies continuity, if continuity fails at (0,0) then differentiability must also fail — regardless of whether partial derivatives exist. This example illustrates that partial derivatives only measure rates of change along the coordinate axes; they say nothing about behavior in diagonal or other directions.

Question 2

Which of the following is a sufficient condition that guarantees a function f(x,y) is differentiable at (a,b)?

• A. Both partial derivatives fₓ and f_y exist at (a,b)
• B. f is continuous at (a,b)
• C. Both partial derivatives fₓ and f_y exist and are continuous in a neighborhood of (a,b)
• D. The gradient vector ∇f(a,b) is nonzero

Answer: (C)

Mere existence of partial derivatives (option A) does not guarantee differentiability — the function could behave badly in non-axis directions. Continuity alone (option B) doesn't imply differentiability either. The reliable sufficient condition is that the partial derivatives *exist and are continuous* in a neighborhood of (a,b). This guarantees the linear approximation works in all directions, not just along the axes.

Question 3

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

• T. True
• F. False

Answer: True

Differentiability implies the existence of partial derivatives. The formal definition of differentiability requires that there exist constants L₁ and L₂ such that the linear approximation using L₁h + L₂k vanishes faster than √(h²+k²). When this holds, restricting to h=0 or k=0 recovers the limit definitions of f_y and fₓ respectively — so both must exist and equal L₂ and L₁. The implication goes one way: differentiability ⟹ partial derivatives exist (but not conversely).

Question 4

If both partial derivatives of f(x,y) exist at most point in ℝ², then f is differentiable at nearly every point in ℝ².

• T. True
• F. False

Answer: False

Existence of partial derivatives does not guarantee differentiability. A classic counterexample: f(x,y) = xy/√(x²+y²) for (x,y) ≠ (0,0) and f(0,0) = 0. Both fₓ(0,0) and f_y(0,0) exist (both equal 0), yet f is not differentiable at the origin because the linear approximation using 0·h + 0·k does not vanish faster than √(h²+k²) when approaching along the line y = x. The partial derivatives only measure axis-aligned rates of change, missing the function's behavior in other directions.

Question 5

Why is the existence of partial derivatives not sufficient to guarantee differentiability in multivariable calculus, when the existence of a derivative is sufficient in single-variable calculus?

Model answer: In single-variable calculus, there is only one direction to approach a point (from the left or right along the number line), so the derivative captures all approach directions. In multivariable calculus, a point in ℝ² can be approached from infinitely many directions. Partial derivatives only measure rates of change along the x-axis and y-axis directions. A function can have well-defined rates along both axes while behaving wildly in diagonal or other directions — for example, not even being continuous. Differentiability requires the linear approximation to work uniformly across all approach directions, which is a genuinely stronger condition.

The asymmetry between 1D and 2D differentiability is one of the deepest conceptual shifts in multivariable calculus. In 1D, 'differentiable' = 'has a derivative' = 'has a tangent line approximation.' In 2D, 'has partial derivatives' is far weaker than 'has a tangent plane approximation.' The definition of differentiability captures exactly what 'tangent plane' should mean: a linear function that approximates f well in *every* direction, not just north-south and east-west.