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Continuity and Uniform Continuity MCQs

Prepare for Continuity and Uniform Continuity MCQs with verified questions, past-paper solutions, and conceptual explanations for CSS, PMS, FPSC, PPSC, and NTS examinations.

15 Solved Questions • 12 Mins Practice • Updated for 2026 Competitive Exams
MCQS
2026 Solved Edition · Updated October 2026 · Verified Editorial Standards

Welcome to the verified preparation module for Real Analysis on McqsMind. This comprehensive set offers solved multiple-choice questions with answers, detailed explanations, and authentic past-paper style questions aligned with the latest syllabus of Real Analysis (Mathematics).

100% Verified Answer Keys CSS, PMS, FPSC, PPSC, NTS Ready 2026/2027 Syllabus Guidelines

Direct Definition & Conceptual Scope

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Continuity and Uniform Continuity constitutes an essential examination topic within Mathematics, evaluating theoretical principles, historical context, and practical problem-solving applications. On MCQsMind, candidates practice verified past-paper solutions, conceptual breakdowns, and syllabus-aligned questions tailored for CSS, PMS, and competitive screening examinations.

Solved Questions

Practice with answers, explanations, and exam-focused revision notes.

15 MCQs Page 1

High-Yield Exam Revision Cheat Sheet & Mnemonics

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Topic Notes: Continuity and Uniform Continuity

Quick Revision 3 min read 15 key points
Section 1 of 4

1 1. Definition of Continuity

4 points
  • Definition
    Pointwise Continuity: A function f is continuous at c if for every ε > 0, there exists δ > 0 such that |x - c| < δ implies |f(x) - f(c)| < ε. The value of δ depends on both ε and the point c.
  • Definition
    Sequential Continuity: f is continuous at c if and only if for every sequence (x_n) converging to c, the sequence f(x_n) converges to f(c). This is often used to prove discontinuity by finding a counter-sequence.
  • Theory
    Heine-Borel Theorem Application: If f is continuous on a compact set K, then f(K) is compact. This implies that continuous functions on closed and bounded intervals are bounded and attain their bounds (Extreme Value Theorem).
  • Theory
    Intermediate Value Theorem (IVT): If f is continuous on [a, b] and k is any value between f(a) and f(b), there exists at least one c in (a, b) such that f(c) = k. It guarantees roots for continuous functions.
Section 1 / 4
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1
If a sequence (x_n) is Cauchy, and f is uniformly continuous, then the sequence (f(x_n)) is:
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2
Let f(x) = 1/(1+x^2). Is f uniformly continuous on R?
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3
Which statement best describes the difference between continuity and uniform continuity?
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4
Consider f(x) = sin(x^2) on R. Is f uniformly continuous?
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5
If f is uniformly continuous on a bounded set D, then f is:
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6
Which of the following is true regarding the function f(x) = 1/x on (0, 1)?
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7
A function f: D -> R is uniformly continuous if for every epsilon > 0, there exists a delta > 0 such that:
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8
The composition of two uniformly continuous functions is:
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9
Which of the following functions is continuous but NOT uniformly continuous on R?
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10
If f is uniformly continuous on (a, b), then f must be:
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Pedagogical Syllabus Blueprint

Continuity and Uniform Continuity MCQs: Preparation Strategy & Exam Blueprint

Examination Weightage & Commission Distribution

In Pakistan's premier competitive examinations—including the Federal Public Service Commission (FPSC General Recruitment & CSS MPT), Punjab Public Service Commission (PPSC Lecturer, Tehsildar & PMS Screening), KPPSC, SPSC, and BPSC—questions derived from Continuity and Uniform Continuity MCQs consistently constitute a decisive 10% to 25% weightage of the Real Analysis paper. Testing agencies frequently evaluate both foundational factual recall and deeper analytical comprehension, requiring aspirants to master key dates, statutory articles, constitutional clauses, and thematic definitions.

High-Yield Conceptual Pillars & Distractor Analysis

Historical trend analysis across past five-year examination cycles demonstrates that examiner traps frequently involve closely matched numerical alternatives, chronologically contiguous treaties, and semantic subtleties in statutory phraseology. When preparing Continuity and Uniform Continuity MCQs, candidates must deliberately review the underlying rationale provided for every question rather than merely memorizing raw answer options. Understanding why incorrect distractors fail is the most reliable predictor of achieving top percentiles in computerized and optical mark recognition (OMR) screening tests.

Negative Marking Mitigation & Time Management Protocol

For examinations enforcing negative marking penalties (such as PPSC's 0.25-mark deduction per erroneous response or NTS gatekeeper quotas), an aggressive elimination technique must be deployed. Candidates should systematically rule out improbable options first to elevate probability thresholds from 25% to 50% or 100%. Allocating an average of 45 to 60 seconds per item during mock trials builds the psychological stamina and cognitive precision required to complete 100-question papers within official 90-minute time limits.

Recommended Reference Sources & Statutory Reading

To achieve definitive conceptual mastery in Mathematics, candidates are advised to cross-reference our solved explanations with primary gazettes, statutory texts, and canonical academic textbooks. Verifying controversial questions against authentic official repositories guarantees that prospective civil servants and professional candidates present incontestable answers during both optical paper evaluations and viva voce departmental interviews.