Continuity Questions and Answers

Practice Continuity Questions and Answers

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Continuity Questions and Answers

Q1. Let f,g:R→R be a continuous function whose graph do not intersect. Then for which function below the graph lies entirely on one side of the X-axis

(a) f
(b) g+f
(c) g-f
(d) g.f

Q2. f(x)=ex-e-x , g(x)=ex+e-x. Then

(a) Both f and g are even functions
(b) Both f and g are odd functions
(c) f is odd, g is even
(d) f is even, g is odd

Q3. Let F:R→R be a monotone function. Then

(a) F has no discontinuities.
(b) F has only finiteily many discontinuities.
(c.) F can have at most countably many discontinuities.
(d.)F can bave uncountably many discontinuities

Q4. Let X be a set of f,g:X→Y be functions. We can say that fog is bijective if

(a) at least one of f , g is bijective
(b) both f and g are bijective
(c) f is one-one and is onto
(d) f is onto and g is one-one

Q5. Let f,g:R→R be functions. We can conclude that h(x) ≤ f(x)∀x∈R if we define h:R→R as

(a) min {g(x), f(x) + g(x)}
(b) min {f(x), f(x) + g(x)}
(c) max {g(x), f(x) + g(x)}
(d) max {f(x), f(x) + g(x)}

Q6. Let X be a non-empty set, f:X→X be a function and let A,B ⊂ X. Then the identity f(A∩B) = f(A)∩f(B) is

(a) Always holds
(b) holds if f is one-one
(c) holds if f is onto
(d) holds if A∪B = X

Q7. The range of the function f(x) = x/√(x2+1) , x∈R is

(a) [-1, 1]
(b) [-1, 1)
(c) (-1, 1]
(d) None of these

Q8. Let f:X→X such that f{f(x)}=x for all x∈X .Then

(a) f is one-to-one and onto
(b) f is one-to-one but not onto
(c) f is onto but not one-to-one
(d) f need not be either one-to-one or onto

Q9. A polynomial of. odd degree with real coefficients must have

(a) at least one real root.
(b) no real root.
(c) only real roots.
(d) at least one root which is not real.

Q10. Consider the following sets of functions on

R,W=The set of constant functions on R ,
X =The set of polynomial functions on R,
Y = The set of continuous functions on R,
Z = The set of all functions on R

Which of these sets has the same cardinality as that of R

(a) Only W
(b) Only W and X
(c) Only , X and Z
(d) Only W, X, Y

Q11. If f:[0,1]→(0,1) is a continuous mapping then which of the following is NOT true?

(a) F⊆[0,1] is a closed set implies f(F) is closed in R.
(b) If f(0)<f(1) then f([0,1]) must be equal to [f(0), f(1)]
(c) There must exist x∈(0,1) such that f(x)=x.
(d) f:([0,1]) ≠ (0,1).

Q12. In which of the following cases, there is no continuous function f from the set S, onto the set T ?

(a) S = [0, 1], T = R
(b) S = (0, 1), T = R
(c) S = (0, 1), T = (0,1]
(d) S = R, T = (0,1)

Q13. Let f:R→R be a strictly increasing continuous function. If {an} is sequence in [0,1]  then the sequence {f(an)} is

(a) Increasing
(b) Bounded
(c) Comvergent
(d) Not necessarily bounded

Q14. Let f:R→R be defined by f(x)= [x2]. The points of discontinuity of f are

(a) Only the integral points
(b) All rational numbers
(c) {±√n : n is.a non-negative integer}
(d) All real numbers

Q15. Which of the following functions has exactly two points of discontinuity ?


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Answer Key (if you find any Question/answer wrong, feel free to correct us.)


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