For f(x, y) = xy/(x² + y²) with f(0, 0) = 0: both iterated limits — first fixing x = 0 then taking y → 0, and vice versa — equal 0. Does this prove f is continuous at the origin?
AYes — if both iterated limits equal the function value, the function is continuous at that point.
BNo — continuity requires the joint limit lim_{(x,y)→(0,0)} to exist and equal f(0,0), and iterated limits don't guarantee the joint limit exists.
CYes — checking along the coordinate axes gives sufficient coverage of all directions approaching the origin.
DNo — continuity in multiple variables requires checking infinitely many paths, which is impossible to verify in practice.
This is the classic counterexample. Along any line y = mx, the limit is m/(1 + m²), which depends on m — so different lines give different limits and the joint limit does not exist. f is therefore not continuous at the origin, despite both iterated limits equaling 0. Option A is the central misconception: passing both axis-direction limits is necessary but not sufficient. Option D is also wrong — finding a single path that gives a different value is sufficient to disprove the limit's existence.
Question 2 Multiple Choice
A student wants to prove that f(x, y) = sin(x² + y²)/(x² + y²), defined as 1 at (0, 0), is continuous at the origin. Which approach is valid?
AShow that the limit as x → 0 (holding y = 0) equals 1, and the limit as y → 0 (holding x = 0) equals 1.
BSubstitute x = r cos θ, y = r sin θ so x² + y² = r², then show the limit as r → 0 equals 1 regardless of θ.
CArgue by symmetry: the function depends only on x² + y², so all directions are equivalent, meaning one direction suffices.
DGraph the function near the origin and observe no visible discontinuity.
Converting to polar coordinates is the standard strategy for this class of problem. Since x² + y² = r², the function becomes sin(r²)/r² → 1 as r → 0, and this limit is independent of θ. Checking all directions simultaneously is what polar coordinates achieve. Option A only checks two specific paths (the axes), which is insufficient — a different path might give a different limit. Option C sounds similar to option B but is informal and would not constitute a proof.
Question 3 True / False
If both partial derivatives ∂f/∂x and ∂f/∂y exist at a point (a, b), the function f should be continuous there.
TTrue
FFalse
Answer: False
This is one of the most surprising results in multivariable calculus. Partial derivatives measure behavior only along the coordinate axes — they say nothing about approaching along other directions. A function can have both partial derivatives at a point while being discontinuous there. For example, f(x, y) = xy/(x² + y²) has both partial derivatives equal to 0 at the origin, yet is discontinuous there. Continuity (and later, full differentiability) requires control over all paths, not just the axial ones.
Question 4 True / False
Checking that lim_{(x,y)→(a,b)} f(x, y) equals f(a, b) along most straight line through (a, b) is sufficient to prove continuity at that point.
TTrue
FFalse
Answer: False
Straight lines through a point don't cover all possible approach paths — curved paths like parabolas y = cx² are not captured. The classic counterexample is f(x, y) = x²y/(x⁴ + y²): along every line y = mx through the origin, the limit is 0, but along the parabola y = x², the limit is 1/2. So the joint limit does not exist and f is discontinuous at the origin, despite passing every straight-line test. Disproving continuity requires only one bad path; proving it requires the ε-δ definition or a technique (like polar coordinates) that controls all paths simultaneously.
Question 5 Short Answer
Why does the concept of continuity become richer — and harder to verify — in multiple variables compared to single-variable calculus, and what is the consequence for how we must test it?
Think about your answer, then reveal below.
Model answer: In single-variable calculus, a point has only two approach directions (left and right), so checking both suffices. In multiple variables, a point can be approached along infinitely many paths — every line, parabola, spiral, etc. — and the limit must be the same along all of them. This means checking any finite set of paths (such as the coordinate axes) can never prove continuity; it can only disprove it by exhibiting a path that gives a different value. Proving continuity usually requires the ε-δ definition or a substitution (like polar coordinates) that controls all directions simultaneously.
The core insight is that 'infinitely many paths' requires fundamentally different proof strategies. The existence of partial derivatives — which only tests the coordinate axes — is particularly insufficient. This gap between partial derivative existence and continuity motivates the stronger concept of differentiability, which requires a linear approximation to hold in all directions.