Practice Continuity Questions and Answers

**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)=e ^{x}-e^{-x} , g(x)=e^{x}+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}/√(x^{2}+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 {a _{n}} is sequence in [0,1] then the sequence {f(a_{n})} is**

(a) Increasing

(b) Bounded

(c) Comvergent

(d) Not necessarily bounded

**Q14. Let f:R→R be defined by f(x)= [x ^{2}]. 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.)

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

01. | (c) | 06. | (b) | 11. | (b) |

02. | (c) | 07. | (d) | 12. | (a) |

03. | (c) | 08. | (a) | 13. | (b) |

04. | (b) | 09. | (a) | 14. | (c) |

05. | (b) | 10. | (d) | 15. | (b) |