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📝 Continuity of multivariable functions (14 MCQs)

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

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What is Continuity of multivariable functions?

Definition:
A function $f$ is continuous at $(a,b)$ if $\lim_{(x,y)\to(a,b)} f(x,y) = f(a,b)$ , requiring defined value, existing limit, and equality.

Example:
$f(x,y) = xy$ is continuous everywhere, but $g(x,y) = \frac{xy}{x^2+y^2}$ (with $g(0,0)=0$ ) is discontinuous at origin due to nonexistent limit.

Reason:
Continuity guarantees no sudden jumps or holes, enabling application of intermediate value theorems and ensuring physical models behave predictably.

Easy

Medium

Hard

📝 All Continuity of multivariable functions MCQs

Q1. A function $f(x,y)$ has partial derivatives $f_x(0,0) = 2$ and $f_y(0,0) = -1$ . Which statement best evaluates the continuity of $f$ at the origin?

A.The function is definitely continuous because both partial derivatives exist and are finite.

B.The function is discontinuous because partial derivatives cannot have opposite signs at a point.

C.Continuity cannot be determined solely from the existence of partial derivatives; further analysis of the limit is required. ✅

D.The function is continuous only if

f(0,0) = 0

, otherwise it is discontinuous.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Existence of partial derivatives at a point does not guarantee continuity in multivariable calculus. A classic counterexample involves functions where partials exist but the function approaches different values along non-linear paths. Students must distinguish between directional rates of change and overall limit behavior to avoid the common misconception that differentiability components imply continuity.

Q2. Consider $f(x,y) = \frac{x^2 y}{x^4 + y^2}$ for $(x,y) \neq (0,0)$ and $f(0,0)=0$ . Along every straight line $y=mx$ , the limit is 0. Why is this insufficient to prove continuity at the origin?

A.Straight lines do not cover the entire domain of the function.

B.The function is actually continuous because all linear paths yield the same limit.

C.Non-linear paths like

y=x^2

may yield a different limit, revealing path-dependence despite linear consistency. ✅

D.The denominator becomes zero along certain curves making the limit undefined.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: This question targets the error analysis of assuming linear path testing is sufficient for multivariable limits. Even if infinitely many straight lines agree, a parabolic or higher-order path can expose discontinuity. True continuity requires the limit to be unique regardless of the approach trajectory, demanding rigorous epsilon-delta verification or identification of a specific counter-path.

Q3. A contour map of $f(x,y)$ shows level curves becoming infinitely dense as they approach point $P$ , with adjacent contours labeled 5 and 10 having no intermediate values between them near $P$ . What does this graphical feature suggest about continuity at $P$ ?

A.The function is continuous but has a very steep gradient at

P

B.The function likely has a jump discontinuity or essential singularity at

P

due to abrupt value changes. ✅

C.The contour density indicates high differentiability and smoothness at

P

D.The graph is inconclusive without seeing the actual surface plot.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Graphical interpretation of contour maps requires understanding that infinite density with discrete value jumps indicates a breakdown in continuity. In continuous functions, level sets should transition smoothly. This visual cue helps students connect abstract limit definitions to geometric representations, distinguishing between steep but continuous gradients and actual discontinuities where intermediate value property fails locally.

Q4. Given $f(x,y) = |x| + |y|$ at the origin, which combination of properties correctly describes the behavior regarding continuity and partial derivatives?

A.Continuous with existing partial derivatives in all directions.

B.Discontinuous with non-existing partial derivatives.

C.Continuous but partial derivatives

f_x

and

f_y

do not exist at the origin. ✅

D.Discontinuous but partial derivatives exist due to symmetry.

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: This direct recall question reinforces that absolute value functions remain continuous everywhere including corners, yet fail to possess partial derivatives at kinks. The distinction is fundamental: continuity concerns limit agreement with function value, while partial derivatives require local linear approximability along axes. Students often confuse these concepts, thinking non-differentiability implies discontinuity in multivariable contexts.

Q5. In modeling heat distribution $T(x,y,t)$ , a sudden material interface causes $T$ to be defined piecewise. If $\lim_{(x,y)\to(a,b)} T(x,y,t_0)$ exists but differs from $T(a,b,t_0)$ , what is the physical implication for the model's validity at that instant?

A.The temperature is continuous but the material conductivity is infinite.

B.The model contains a removable discontinuity suggesting an idealized point source or measurement error at the interface. ✅

C.Heat flow is impossible at that point due to thermal equilibrium.

D.The partial derivatives with respect to space are zero at the interface.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Application questions link mathematical continuity to physical modeling realism. A removable discontinuity in temperature suggests either an idealization artifact or data inconsistency rather than genuine physical behavior. Recognizing this allows engineers to refine models by redefining the point value or investigating measurement limitations, demonstrating how mathematical classification guides practical problem-solving in applied multivariable scenarios.

Q6. Student A claims $f(x,y) = \frac{xy}{\sqrt{x^2+y^2}}$ is discontinuous at origin because polar substitution gives $r\cos\theta\sin\theta$ which depends on $\theta$ . Identify the flaw in this reasoning.

A.Polar coordinates cannot be used for continuity testing.

B.The expression actually simplifies to a form bounded by

r

, making the limit zero independent of

\theta

r\to 0

. ✅

C.The student forgot to check Cartesian paths first.

D.The function is indeed discontinuous; the reasoning is correct.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Error analysis here addresses misinterpretation of polar forms. While $\cos\theta\sin\theta$ varies with angle, multiplication by $r$ forces the entire expression toward zero uniformly. The squeeze theorem applies because trigonometric factors are bounded. This multi-step reasoning corrects the misconception that any angular dependence implies path-dependence, emphasizing the role of radial decay in establishing continuity.

Q7. Compare two functions at the origin: $f(x,y)=\frac{x^3}{x^2+y^2}$ and $g(x,y)=\frac{x^2}{x^2+y^2}$ . Both have well-defined partial derivatives along axes. Which accurately characterizes their continuity difference?

A.Both are continuous because partials exist.

B.Neither is continuous due to denominator vanishing.

f

is continuous due to higher-degree numerator dominating;

g

is discontinuous as limit depends on approach path. ✅

g

is continuous but

f

is not due to odd power asymmetry.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Mixed concept comparison requires analyzing degree homogeneity. Function $f$ has numerator degree exceeding denominator effectively after simplification, enabling squeeze theorem application. Function $g$ has equal degrees yielding path-dependent limits. This contrasts how algebraic structure determines continuity despite similar partial derivative existence, reinforcing that continuity depends on global limit behavior rather than axial properties alone.

Q8. If $f(x,y)$ is continuous at $(a,b)$ and $g(t)$ is continuous at $t=f(a,b)$ , which statement about the composition $h(x,y)=g(f(x,y))$ represents the strongest valid conclusion?

h

is continuous at

(a,b)

only if

f

is also differentiable there.

h

is always continuous at

(a,b)

by the composition theorem for multivariable functions. ✅

h

is continuous only if

g

is uniformly continuous.

D.Continuity of

h

cannot be guaranteed without knowing the partial derivatives of

f

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Conceptual understanding of composition rules is essential. The multivariable composition theorem states that continuity is preserved under composition without requiring differentiability. This distinguishes continuity from stronger properties like differentiability. Students must recognize that basic topological properties transfer through continuous mappings, avoiding over-complication by unnecessarily invoking derivative conditions when only continuity is at stake.

Q9. A surface plot shows $f(x,y)$ approaching 3 along all visible paths toward origin, yet $f(0,0)=5$ . Numerical sampling within $\delta=0.001$ consistently yields values near 3. How should one classify this discontinuity?

A.Essential discontinuity due to oscillatory behavior.

B.Jump discontinuity because left and right limits differ.

C.Removable discontinuity since the limit exists but differs from the defined value. ✅

D.Infinite discontinuity due to unbounded growth near origin.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Scenario-based classification requires interpreting numerical and graphical evidence together. Consistent approach to a single value distinct from the function definition characterizes removable discontinuities. This differs from jump or essential types. Recognizing this pattern enables correction by redefining $f(0,0)=3$ , illustrating how computational exploration supports theoretical classification in practical analysis of multivariable functions.

Q10. For $f(x,y) = \begin{cases} \frac{x^2-y^2}{x^2+y^2} & (x,y)\neq(0,0) \\ c & (x,y)=(0,0) \end{cases}$ , can any choice of $c$ make $f$ continuous at the origin?

A.Yes, choosing

c=0

ensures continuity.

B.Yes, choosing

c=1

matches the limit along the x-axis.

C.No, because the limit does not exist as different paths yield different values. ✅

D.No, because the function is unbounded near the origin.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: This application question tests understanding of necessary conditions for continuity. Since approaching along $y=0$ gives 1 and along $x=0$ gives -1, no single limit exists. Therefore no value of $c$ can satisfy the continuity definition. This reinforces that defining a point value cannot repair fundamental path-dependence, distinguishing removable from essential discontinuities.

Q11. When verifying continuity of $f(x,y)=\sqrt{|xy|}$ at origin using epsilon-delta, which inequality chain provides the most efficient bounding strategy?

|f(x,y)-0| \leq |x|+|y|

via triangle inequality.

|f(x,y)| \leq \sqrt{x^2+y^2}

using AM-GM or direct comparison. ✅

|f(x,y)| \leq x^2+y^2

by squaring both sides.

|f(x,y)| \leq |xy|

since square root reduces magnitude.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Multi-step reasoning in epsilon-delta proofs requires optimal bounding. Using $|xy| \leq \frac{x^2+y^2}{2}$ leads to $\sqrt{|xy|} \leq \sqrt{\frac{x^2+y^2}{2}}$ , directly relating to norm. This demonstrates strategic inequality selection over brute force. Efficient bounds simplify delta selection and reveal the function's Hölder continuity nature, showcasing advanced technique beyond basic limit computation.

Q12. An Olympiad-style challenge: Let $f:\mathbb{R}^2\to\mathbb{R}$ satisfy $|f(x,y)-f(u,v)| \leq K\sqrt{|x-u|+|y-v|}$ for constant $K>0$ . What is the strongest continuity property guaranteed at every point?

A.Merely pointwise continuity without uniform control.

B.Uniform continuity and Hölder continuity with exponent 1/2, implying continuity but not necessarily differentiability. ✅

C.Lipschitz continuity ensuring differentiability almost everywhere.

D.Absolute continuity guaranteeing integrability of partial derivatives.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: This challenging problem connects metric inequalities to continuity classes. The given condition defines Hölder continuity with exponent 1/2, which is stronger than ordinary continuity but weaker than Lipschitz. Such functions are uniformly continuous but may lack differentiability anywhere. Recognizing this hierarchy requires synthesizing analysis concepts beyond standard curriculum, testing deep understanding of continuity moduli and their implications.

Q13. In error analysis of numerical methods, approximating $f(x,y)$ near a suspected discontinuity yields oscillating values between 2 and 4 as grid refines. Partial derivatives computed numerically grow without bound. What diagnosis is most consistent?

A.The function is continuous but highly oscillatory with large derivatives.

B.The function has an essential discontinuity where the limit does not settle to any value. ✅

C.Numerical instability due to floating-point errors rather than mathematical behavior.

D.The function is removable discontinuous with poorly chosen sample points.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Interpreting computational artifacts requires distinguishing numerical noise from genuine mathematical pathology. Unbounded derivative growth combined with persistent oscillation across refinements signals essential discontinuity rather than mere steepness. This scenario-based diagnosis integrates numerical analysis with theoretical continuity concepts, emphasizing that algorithmic behavior can serve as diagnostic evidence when analytical methods are inconclusive or computationally intensive.

Q14. Which statement correctly identifies a subtle flaw in claiming $f_x$ and $f_y$ continuous in neighborhood implies $f$ continuous at point"?"

A.The implication is actually valid; there is no flaw.

B.Continuity of partials guarantees differentiability, which implies continuity, so the logic holds.

C.The statement assumes partials exist at the point itself, but neighborhood continuity alone doesn't ensure existence at the point. ✅

D.Partial derivatives being continuous is irrelevant to function continuity.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: This conceptual question probes precise logical dependencies. While continuous partials in a neighborhood typically imply differentiability (hence continuity) at interior points, the technical requirement includes existence at the point. Overlooking this subtlety reflects incomplete understanding of theorem hypotheses. Rigorous analysis demands verifying all conditions, preventing overgeneralization of sufficient conditions in multivariable continuity theory.

🔗 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

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