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📝 Continuity at boundary points in multivariable (12 MCQs)

📖 From Calculus • 14. Partial Derivatives Calculus • 12 questions available

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What is Continuity at boundary points in multivariable?

Definition:
Continuity at boundary point $P$ of domain $D$ requires $\lim_{(x,y)\to P, (x,y)\in D} f(x,y) = f(P)$ , considering only approaches within $D$ .

Example:
$f(x,y) = \sqrt{1-x^2-y^2}$ is continuous at boundary point $(1,0)$ when restricted to domain $x^2+y^2 \leq 1$ .

Reason:
Many optimization problems occur on boundaries; restricting approach direction makes continuity achievable even when full-space limits would not exist.

Easy

Medium

Hard

📝 All Continuity at boundary points in multivariable MCQs

Q1. A function $f(x,y)$ is defined on the closed disk $x^2 + y^2 \leq 1$ . At the boundary point $(1,0)$ , the limit along every straight line path within the domain equals 5. Which statement best evaluates the continuity of $f$ at this point?

A.The function is definitely continuous because all linear paths agree.

B.The function is discontinuous because parabolic paths were not tested.

C.Continuity cannot be confirmed without verifying the limit for all possible curves approaching within the domain. ✅

D.The function is continuous only if the partial derivatives exist at the boundary.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Agreement along straight lines is insufficient for multivariable continuity at boundary points. Students often mistakenly apply single-variable intuition or assume linearity suffices. True continuity requires the limit to be unique regardless of the approach path, including non-linear curves constrained within the domain's geometry.

Q2. Consider $f(x,y) = \frac{x^2 y}{x^4 + y^2}$ for $(x,y) \neq (0,0)$ and $f(0,0)=0$ , restricted to the domain $y \geq 0$ . A student claims $f$ is continuous at the origin because limits along $y=mx$ are zero. What is the flaw in this reasoning?

A.The student failed to check vertical paths.

B.The domain restriction

y \geq 0

invalidates the use of any linear paths.

C.Linear paths yield zero, but the path

y=x^2

yields a non-zero limit, disproving continuity. ✅

D.Partial derivatives do not exist, which implies discontinuity.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: This error analysis question targets the common misconception that checking linear paths proves continuity. Even with domain restrictions, non-linear paths like $y=x^2$ can reveal different limiting behavior. Continuity demands uniform convergence from all valid directions, making multi-path testing essential for rigorous verification at boundary points.

Q3. A temperature model $T(x,y)$ on a rectangular plate $[0,1] \times [0,1]$ has a known interior formula but measured boundary values. If $\lim_{(x,y) \to (0,0)^+} T(x,y) = 25$ from the interior, but the sensor at corner $(0,0)$ reads 30, what does this imply physically and mathematically?

A.The sensor is faulty; mathematical limits always override measurements.

B.There is a thermal singularity or insulation failure at the corner, indicating mathematical discontinuity matching physical reality. ✅

C.The function is continuous because one-sided limits exist in the first quadrant.

D.Boundary points cannot be evaluated for continuity in applied models.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: This scenario-based question links mathematical discontinuity to physical phenomena. A mismatch between interior limit and boundary value indicates a genuine discontinuity, not just measurement error. Understanding this connection helps students appreciate why boundary continuity matters in modeling heat transfer, fluid dynamics, and other real-world applications involving partial derivatives.

Q4. Given a contour plot of $f(x,y)$ on $x \geq 0$ , where level curves near the boundary $x=0$ become increasingly dense and cluster toward different values depending on $y$ -position, what can be inferred about continuity at boundary points?

A.The function is continuous because contours exist everywhere.

B.Dense contours indicate high differentiability, implying continuity.

C.The clustering suggests the limit depends on the approach direction along the boundary, indicating potential discontinuity. ✅

D.Contour plots cannot provide information about boundary continuity.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Graph interpretation questions require translating visual density patterns into analytical conclusions. Clustering level curves near a boundary signal that function values change rapidly and may not converge uniformly. This visual cue corresponds to non-existent or path-dependent limits, helping students connect graphical intuition with formal epsilon-delta definitions of continuity at edge points.

Q5. For $f(x,y) = \sqrt{x} \sin(y/x)$ defined on $x > 0$ extended to $x=0$ by $f(0,y)=0$ , analyze continuity at boundary point $(0,0)$ . Which combination of concepts is required for proper evaluation?

A.Single-variable limit laws and derivative existence.

B.Squeeze theorem application recognizing

|\sin| \leq 1

combined with domain-restricted approach paths. ✅

C.Polar coordinate transformation alone is sufficient.

D.Checking partial derivatives at the origin determines continuity.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: This mixed-concepts problem requires combining the squeeze theorem with awareness of domain constraints. Since $x>0$ , approaches are restricted to right half-plane. The bounded sine term multiplied by vanishing $\sqrt{x}$ forces the limit to zero via squeezing, demonstrating how inequality bounds can establish continuity even when direct substitution fails at boundaries.

Q6. A student argues that since $f_x$ and $f_y$ exist at boundary point $(a,b)$ of domain $D$ , then $f$ must be continuous there. Evaluate this claim using counterexample logic.

A.The claim is correct; partial derivatives guarantee continuity everywhere.

B.The claim is false; consider

f(x,y)=1

x=0

y=0

, else 0, at origin on first quadrant. ✅

C.Existence of partials implies continuity only for polynomial functions.

D.Boundary points automatically satisfy continuity by definition.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: This error analysis targets the persistent misconception linking partial derivative existence to continuity. The provided counterexample shows a function with existing partials (both zero) yet clearly discontinuous. Students must understand that partial derivatives examine only axial behavior, while continuity requires convergence from all directions within the domain, especially critical at boundary regions.

Q7. In optimizing profit $P(x,y)$ over feasible region $x \geq 0, y \geq 0$ , you find an interior critical point and boundary candidates. Why must continuity at boundary corners be verified before applying extreme value theorem guarantees?

A.Extreme value theorem applies regardless of continuity on closed domains.

B.Discontinuity at corners could mean global extrema occur at undefined limits rather than computed values. ✅

C.Corners are never optimal solutions in economic models.

D.Only differentiability matters for optimization, not continuity.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: This application question connects continuity verification to optimization validity. The extreme value theorem requires continuity on compact sets. If boundary corners have discontinuities, maximum/minimum might not be attained at calculated points. Students must recognize that theoretical guarantees depend on foundational hypotheses, making continuity checks at domain edges essential for reliable decision-making in constrained optimization problems.

Q8. When extending $f(x,y) = \frac{xy}{\sqrt{x^2+y^2}}$ to the origin on domain $x^2+y^2 \leq 4$ , which method most efficiently establishes continuity at this boundary-adjacent point while avoiding computational pitfalls?

A.Direct substitution after rationalizing the denominator.

B.Testing multiple specific paths including axes and diagonals.

C.Converting to polar coordinates to show limit equals zero independent of angle. ✅

D.Computing both partial derivatives and showing they equal zero.

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Polar coordinates elegantly handle radial symmetry near origin, transforming two-variable limit into single-variable analysis. Since $r \to 0$ uniformly regardless of $\theta$ , and expression simplifies to $r \cos\theta \sin\theta$ , boundedness ensures convergence. This method avoids path-testing incompleteness while respecting domain constraints, demonstrating efficient technique selection for boundary-adjacent continuity verification.

Q9. Two functions $f$ and $g$ are continuous on open set $U$ but only $f$ extends continuously to boundary $\partial U$ . For composite $h = f \circ g$ , what additional condition ensures $h$ extends continuously to boundary points where $g$ maps into $\partial U$ ?

A.No additional conditions needed; composition preserves continuity.

g

must map boundary points to interior points where

f

is defined.

f

must be uniformly continuous on the range of

g

near boundary.

g

must itself extend continuously to those boundary points. ✅

💡 Difficulty: easy | ✅ Correct: D

📖 Explanation: This Olympiad-style question probes deep understanding of function composition at boundaries. Even if $f$ extends continuously, $h$ fails if $g$ lacks boundary extension, as composition requires well-defined inputs. Students must trace dependency chains: boundary behavior of outer function is irrelevant if inner function doesn't reach appropriate domain points, revealing subtle topological constraints.

Q10. A piecewise function defines $f(x,y)=x^2$ for $y>0$ and $f(x,y)=y$ for $y \leq 0$ on domain $[-1,1]^2$ . At boundary segment $y=0$ , which analytical approach correctly assesses continuity?

A.Check if left and right derivatives match at each point.

B.Verify that

\lim_{y \to 0^+} x^2 = \lim_{y \to 0^-} y

for each fixed

x

. ✅

C.Since both pieces are polynomials, continuity is automatic.

D.Evaluate partial derivatives from above and below separately.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: This conceptual question emphasizes that boundary continuity for piecewise functions requires matching limits across the interface for each parameter value. Simply having smooth pieces isn't sufficient; the transition must be seamless pointwise. Students often confuse differentiability with continuity or assume polynomial pieces guarantee global continuity, overlooking the need for explicit interface verification at every boundary location.

Q11. In numerical simulation of $f(x,y)$ on triangular domain, grid refinement near hypotenuse boundary shows oscillating approximations despite smooth interior behavior. What does this suggest about theoretical continuity assumptions?

A.Numerical methods always fail near boundaries regardless of theory.

B.The function likely satisfies continuity but lacks uniform continuity near boundary.

C.Oscillations indicate probable discontinuity or unbounded derivative at boundary violating simulation stability. ✅

D.Grid resolution is simply insufficient; no theoretical implication exists.

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: This modeling question bridges computational artifacts with theoretical analysis. Persistent oscillations under refinement typically signal violated regularity assumptions like continuity or bounded variation at boundaries. While numerical issues can arise independently, systematic non-convergence suggests underlying mathematical pathology. Students learn to interpret simulation warnings as diagnostic tools for validating theoretical premises about boundary behavior in applied contexts.

Q12. For $f(x,y) = \frac{x^3}{x^2+y^2}$ on closed unit disk, compare evaluating continuity at boundary point $(1,0)$ versus origin. Which distinction is most pedagogically significant?

A.Both points require identical polar coordinate analysis.

B.Origin needs special handling due to indeterminate form, while

(1,0)

allows direct substitution confirming continuity. ✅

C.Boundary points are inherently more complex than interior points.

D.Neither point presents continuity challenges for rational functions.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: This comparison highlights contextual differences in boundary analysis. At $(1,0)$ , denominator is nonzero allowing direct evaluation, whereas origin requires limit techniques despite being interior. Students often overgeneralize boundary difficulty or underestimate interior singularities. Recognizing when standard evaluation suffices versus when advanced methods are needed develops nuanced judgment essential for efficient problem-solving across diverse domain geometries.

🔗 Related Topics (MCQs)

📝 Functions of Two or More Variables MCQs 📝 Multivariable function notation and terminology MCQs 📝 Functions of two variables from tables MCQs 📝 Level curves and contour plots MCQs 📝 Graphing functions of two variables MCQs

←Previous Topic:Multivariable limits at discontinuities📝 Continuity at boundary points in multivariableFunctions of three variables limits continuity:Next Topic→