Continuity of functions MCQs With Answer

Introduction:

Mastering continuity of functions is essential for B. Pharm students, especially when interpreting drug concentration-time profiles, pharmacokinetic curves, and dose–response graphs. This set of “Continuity of functions MCQs With Answer” covers limits, continuous and discontinuous behavior, removable, jump and infinite discontinuities, intermediate value theorem, and practical examples relevant to pharmaceutical analysis. Each multiple-choice question is crafted to strengthen problem-solving skills and conceptual clarity, with clear answers to aid quick revision. The focused practice helps bridge theoretical calculus concepts with real-world pharmacological data interpretation and modeling. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the formal condition for a function f to be continuous at a point a?

• The derivative at a exists and is finite
• The left-hand and right-hand limits at a exist but are not equal
• lim x→a f(x) exists and equals f(a)
• f(a) is undefined but limits exist

Correct Answer: lim x→a f(x) exists and equals f(a)

Q2. A removable discontinuity at x = a occurs when:

• Left and right limits at a are infinite
• lim x→a f(x) exists but f(a) is different or undefined
• Left and right limits at a are finite but unequal
• f has vertical asymptote at a

Correct Answer: lim x→a f(x) exists but f(a) is different or undefined

Q3. Which of the following describes a jump discontinuity?

• Both one-sided limits are finite but not equal
• Both one-sided limits equal and equal f(a)
• Limits approach infinity on either side
• Function oscillates without limit near a

Correct Answer: Both one-sided limits are finite but not equal

Q4. Infinite (essential) discontinuity is characterized by:

• Function value equals limit at that point
• One-sided limits go to ±∞ or do not exist due to blow-up
• Left and right limits are equal finite numbers
• Limit exists but function is undefined

Correct Answer: One-sided limits go to ±∞ or do not exist due to blow-up

Q5. Which functions are continuous for all real x?

• Rational functions with nonzero denominators
• Polynomials
• Logarithmic functions on all reals
• Square root function on negative reals

Correct Answer: Polynomials

Q6. A rational function is continuous at x = a when:

• The numerator is zero at a
• The denominator is zero at a
• Denominator ≠ 0 at a
• The function is piecewise-defined

Correct Answer: Denominator ≠ 0 at a

Q7. If f and g are continuous at a, which of these is true?

• f + g, f·g are continuous at a
• Only f + g is continuous; product is not
• Neither sum nor product is continuous
• Only product is continuous; sum is not

Correct Answer: f + g, f·g are continuous at a

Q8. Composition rule: If f is continuous at a and g is continuous at f(a), then:

• g∘f may be discontinuous at a
• g∘f is continuous at a
• Only g is continuous everywhere
• f must be differentiable at a

Correct Answer: g∘f is continuous at a

Q9. The Intermediate Value Theorem states that if f is continuous on [a,b], then f attains:

• Only rational values between f(a) and f(b)
• Every value between f(a) and f(b)
• Only integer values between f(a) and f(b)
• No values between f(a) and f(b) necessarily

Correct Answer: Every value between f(a) and f(b)

Q10. A continuous function on a closed interval [a,b] necessarily:

• Has no maxima or minima
• Is differentiable everywhere on [a,b]
• Attains both a maximum and a minimum on [a,b]
• Is unbounded on [a,b]

Correct Answer: Attains both a maximum and a minimum on [a,b]

Q11. Differentiability at a point implies:

• Continuity at that point
• Discontinuity at that point
• Not defined behavior of continuity
• Jump discontinuity at that point

Correct Answer: Continuity at that point

Q12. Continuity at a point does NOT guarantee:

• Existence of left and right limits
• Function value is defined
• Differentiability at that point
• Equality of limit and function value

Correct Answer: Differentiability at that point

Q13. For piecewise function f(x)= {x^2 for x≤1, 2x for x>1}, f is continuous at x=1?

• Yes, because left and right limits equal 1
• No, because left limit 1 and right limit 2 are different
• Yes, because both formulas give same rule
• No, because function is undefined at 1

Correct Answer: No, because left limit 1 and right limit 2 are different

Q14. A function f has lim x→a^- f(x)=3 and lim x→a^+ f(x)=3 but f(a)=5. This is:

• A jump discontinuity
• An infinite discontinuity
• A removable discontinuity
• Continuity

Correct Answer: A removable discontinuity

Q15. The left-hand limit notation is written as:

• lim x→a+ f(x)
• lim x→a^- f(x)
• lim x→∞ f(x)
• lim x→a f(x)

Correct Answer: lim x→a^- f(x)

Q16. A function that oscillates increasingly near a point with no approaching value has:

• Removable discontinuity
• Jump discontinuity
• Oscillatory (essential) discontinuity
• Continuous extension

Correct Answer: Oscillatory (essential) discontinuity

Q17. Which of the following is continuous at x=0 after suitable definition?

• f(x)=sin x/x defined as is
• f(x)=sin x/x with f(0)=1
• f(x)=1/x with f(0)=0
• f(x)=tan x

Correct Answer: f(x)=sin x/x with f(0)=1

Q18. The absolute value function f(x)=|x| is:

• Not continuous at 0
• Continuous and differentiable at 0
• Continuous everywhere but not differentiable at 0
• Discontinuous for x<0

Correct Answer: Continuous everywhere but not differentiable at 0

Q19. If lim x→a f(x) does not exist but one-sided limits are finite and unequal, this indicates:

• Continuity
• Jump discontinuity
• Removable discontinuity
• Uniform continuity

Correct Answer: Jump discontinuity

Q20. Which class of functions is continuous on their entire domain: exp(x), ln(x), sqrt(x), sin(x)?

• All are continuous on R
• exp(x) and sin(x) are continuous on their domains
• ln(x) is continuous on all reals including negatives
• sqrt(x) is continuous on all reals

Correct Answer: exp(x) and sin(x) are continuous on their domains

Q21. A function continuous on a closed bounded interval [a,b] is always:

• Uniformly continuous on [a,b]
• Uniformly continuous only if differentiable
• Not uniformly continuous
• Discontinuous at endpoints

Correct Answer: Uniformly continuous on [a,b]

Q22. Which statement about the inverse of a continuous strictly monotone function on an interval is true?

• Inverse need not exist
• Inverse exists and is continuous on its domain
• Inverse is always differentiable
• Inverse is discontinuous everywhere

Correct Answer: Inverse exists and is continuous on its domain

Q23. A function f is continuous at isolated point a. Which is true?

• f must be discontinuous at isolated points
• f is always continuous at isolated points by definition
• Left and right limits are required at isolated points
• Continuity cannot be defined at isolated points

Correct Answer: f is always continuous at isolated points by definition

Q24. The sequential characterization of continuity says f is continuous at a iff:

• For every sequence x_n→a, f(x_n)→f(a)
• For some sequence x_n→a, f(x_n)→f(a)
• For sequences x_n that do not converge, f(x_n)→f(a)
• Only for monotone sequences

Correct Answer: For every sequence x_n→a, f(x_n)→f(a)

Q25. Which of the following graphs indicates continuity?

• Graph with a hole but filled value different from limit
• Graph with a vertical asymptote
• Graph can be drawn without lifting pen over the interval
• Graph oscillating infinitely at a point

Correct Answer: Graph can be drawn without lifting pen over the interval

Q26. For f(x)= (x^2-1)/(x-1), the point x=1 is:

• A vertical asymptote
• A removable discontinuity that can be filled by f(1)=2
• A removable discontinuity that can be filled by f(1)=1
• A jump discontinuity

Correct Answer: A removable discontinuity that can be filled by f(1)=2

Q27. Which function is discontinuous everywhere?

• f(x)=x
• Dirichlet function: 1 for rationals, 0 for irrationals
• sin x
• e^x

Correct Answer: Dirichlet function: 1 for rationals, 0 for irrationals

Q28. Heaviside step function H(x) is discontinuous at:

• Every x
• x=0 only
• No points; it is continuous everywhere
• x=1 only

Correct Answer: x=0 only

Q29. If lim x→a^- f(x)=L and lim x→a^+ f(x)=L, then:

• lim x→a f(x) does not exist
• lim x→a f(x)=L
• Function must be discontinuous at a
• Limits are infinite

Correct Answer: lim x→a f(x)=L

Q30. A function that is continuous on (−∞,∞) but not differentiable at a finite number of points is:

• Impossible
• Common, e.g., absolute value function
• Always discontinuous
• Not integrable

Correct Answer: Common, e.g., absolute value function

Q31. The epsilon-delta definition of limit is primarily used to:

• Compute numerical limits only
• Formally prove continuity or limit statements
• Graph functions
• Approximate derivatives

Correct Answer: Formally prove continuity or limit statements

Q32. If f is continuous on [a,b] and f(a) and f(b) have opposite signs, then:

• f has no root in (a,b)
• There is at least one c in (a,b) with f(c)=0
• f must be differentiable on (a,b)
• f is discontinuous at some point in (a,b)

Correct Answer: There is at least one c in (a,b) with f(c)=0

Q33. The operation f(x)/g(x) is continuous at a if:

• Both f and g are discontinuous at a
• f and g are continuous at a and g(a)≠0
• g(a)=0 but limit exists
• Only f is continuous at a

Correct Answer: f and g are continuous at a and g(a)≠0

Q34. In pharmacokinetics, a concentration-time curve with a sudden vertical jump likely represents:

• Continuous elimination
• Instantaneous bolus injection causing a discontinuity
• Steady-state absorption
• Mathematically smooth absorption

Correct Answer: Instantaneous bolus injection causing a discontinuity

Q35. Which statement is true about limits and function values for continuity?

• Limit must exist but need not equal function value
• Function value must exist but limit may not
• Both limit must exist and equal the function value
• Neither limit nor function value is important

Correct Answer: Both limit must exist and equal the function value

Q36. Which function has a removable discontinuity at x=0?

• f(x)=sin(1/x)
• f(x)=1/x
• f(x)=(1-cos x)/x^2 with appropriate definition at 0
• f(x)=tan x

Correct Answer: f(x)=(1-cos x)/x^2 with appropriate definition at 0

Q37. For continuous f, if f(a)≠0, then there exists a neighborhood around a where:

• f changes sign infinitely often
• f remains of the same sign as f(a)
• f is undefined
• f has jump discontinuities

Correct Answer: f remains of the same sign as f(a)

Q38. The function f(x)=x^2 is uniformly continuous on:

• All of R
• Any closed bounded interval [a,b]
• (0,∞) only
• Nowhere

Correct Answer: Any closed bounded interval [a,b]

Q39. If f is continuous on (a,b) and has no discontinuities, it must be:

• Constant
• Differentiable
• Continuous by hypothesis; nothing more guaranteed
• Unbounded

Correct Answer: Continuous by hypothesis; nothing more guaranteed

Q40. A function that is continuous everywhere except countably many points could be:

• A monotone function
• The Dirichlet function
• Sin x
• An exponential function

Correct Answer: A monotone function

Q41. Which of these is a method to remove a removable discontinuity?

• Redefine function value at the point to equal the limit
• Introduce a vertical asymptote
• Make left and right limits unequal
• Force the denominator to zero

Correct Answer: Redefine function value at the point to equal the limit

Q42. Continuous functions map connected sets to:

• Disconnected sets
• Connected sets
• Only intervals of integers
• Singleton sets only

Correct Answer: Connected sets

Q43. Which of the following ensures continuity at an endpoint a of [a,b]?

• Existence of two-sided limit at a
• Existence of right-hand limit at a equal to f(a)
• Left-hand limit at a equals f(a)
• Neither one-sided limit matters

Correct Answer: Existence of right-hand limit at a equal to f(a)

Q44. The function f(x)=sin(1/x) for x≠0 and f(0)=0 is:

• Continuous at 0
• Discontinuous at 0 due to oscillation
• Has a removable discontinuity at 0
• Infinite at 0

Correct Answer: Discontinuous at 0 due to oscillation

Q45. Which is a correct consequence of continuity for integrals?

• Continuous functions on [a,b] are not integrable
• Every continuous function on [a,b] is Riemann integrable
• Integration does not use continuity
• Continuous functions have no area under the curve

Correct Answer: Every continuous function on [a,b] is Riemann integrable

Q46. The function f(x)=x^3 is differentiable everywhere. Therefore it is:

• Discontinuous somewhere
• Continuous everywhere
• Only continuous at integers
• Not integrable

Correct Answer: Continuous everywhere

Q47. Which example demonstrates a jump discontinuity in a pharmacological dosage graph?

• Continuous slow infusion
• Immediate bolus raising concentration abruptly
• Exponential decay of drug
• Continuous oral absorption curve

Correct Answer: Immediate bolus raising concentration abruptly

Q48. If f is continuous and nonzero at a, which of the following is true for 1/f near a?

• 1/f is discontinuous near a
• 1/f is continuous near a
• 1/f is undefined for all x
• 1/f must have a jump at a

Correct Answer: 1/f is continuous near a

Q49. For the piecewise function f(x)= {x+1 for x<0, x^2 for x≥0}, f is continuous at x=0 if:

• Left limit 1 and right limit 0 are equal
• Left limit 1 equals right limit 0 so false
• Left limit 1 equals right limit 0 only if adjusted
• Left limit 1 and right limit 0 are different, so not continuous

Correct Answer: Left limit 1 and right limit 0 are different, so not continuous

Q50. Extending a function by defining value at a point to match the limit is called:

• Creating an essential discontinuity
• Continuous extension or removable extension
• Introducing a vertical asymptote
• Making the function oscillatory

Correct Answer: Continuous extension or removable extension

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