This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Continuity”.

1. What is the mathematical expression for the definition of continuity?

a) lim_{x→c}f(x) = f(c) ∀ c ∈ a

b) lim_{x→c}f(x) = f(c) ∀ c ∈ (a,b)

c) lim_{x→c}f(x) = f(c) ∀ c ∈ b

d) lim_{x→a}f(x) = f(c) ∀ c ∈ (a,b)

View Answer

Explanation: A function f defined on (a,b) is said to be continuous on (a,b) if it is continuous at every point of (a,b) i.e., if lim

_{x→c}f(x)=f(c) ∀ c ∈ (a,b).

2. What is the mathematical expression for f is continuous on (a,b)?

a) lim_{x→c}f(x) = f(c) ∀ c ∈ a

b) lim_{x→c}f(x) = f(c) ∀ c ∈ (a,b)

c) lim_{x→c}f(x) = f(c) ∀ c ∈ b

d) lim_{x→a}f(x) = f(c) ∀ c ∈ (a,b)

View Answer

Explanation: A function f defined on (a,b) is said to be continuous on (a,b) if it is continuous at every point of (a,b) i.e., if lim

_{x→c}f(x)=f(c) ∀ c ∈ (a,b).

3. What is the mathematical expression for f is right continuous on (a,b)?

a) lim_{x→a+}f(x)=f(a)

b) lim_{x→a+}f(x)=f(b)

c) lim_{x→b+}f(x)=f(a)

d) lim_{x→a-}f(x)=f(a)

View Answer

Explanation: A function is said to be continuous when it is both left continuous and right continuous. Mathematical expression for a function f is right continuous on (a,b) is lim

_{x→a+}f(x)=f(a).

4. What is the mathematical expression for f is left continuous on (a,b)?

a) lim_{x→a-}f(x)=f(a)

b) lim_{x→b-}f(x)=f(b)

c) lim_{x→a+}f(x)=f(b)

d) lim_{x→b+}f(x)=f(b)

View Answer

Explanation: A function is said to be continuous when it is both left continuous and right continuous. Mathematical expression for a function f is left continuous on (a,b) is lim

_{x→a+}f(x)=f(a).

5. f(x) = c ∀ x ∈ R is continuous on R for a fixed c ∈ R.

a) False

b) True

View Answer

Explanation: If a ∈ R then f(a) = c

lim

_{x→a}f(x)=lim

_{x→a}c=c=f(a)

Hence, f(x) is continuous at any point a ∀ R.

6. What are the kinds of discontinuity?

a) Minor and major kinds

b) Increment and decrement kinds

c) First and second kinds

d) Zero and one kinds

View Answer

Explanation: Kinds of discontinuity are classified as follows.

i. Discontinuity of the first kind: Removable and jump discontinuities.

ii. Discontinuity of the second kind: Oscillating and infinite discontinuities.

7. Is f(x) = \( \begin{cases}

\frac {sin2x}{x} & if \, x \neq 0 \\

1 & if \, x = 0 \\

\end{cases} \) a continuous function?

a) Only in some cases

b) Cannot be determined

c) Continuous

d) Not continuous

View Answer

Explanation: lim

_{x→0}f(x)=lim

_{x→0}\(\frac {sin2x}{x}\) x 2

= 1 x 2

= 2 ≠ f(0)

Hence, given function is not continuous at x = 0

8. What is/are conditions for a function to be continuous on (a,b)?

a) The function is continuous at each point of (a,b)

b) The function is right continuous

c) The function is left continuous

d) Right continuous, left continuous, continuous at each point of (a,b)

View Answer

Explanation: The three conditions required for a function f is said to be continuous on (a,b) if f is continuous at each point of (a,b), f is right continuous at x = a, f is left continuous at x = b.

9. lim_{x→a+}f(x)=f(a) then f(x) is right continuous at x = a.

a) True

b) False

View Answer

Explanation: A function is said to be continuous when it is both left continuous and right continuous. Mathematical expression for a function f is left continuous on (a,b) is lim

_{x→a+}f(x)=f(a).

10. lim_{x→a-}f(x)=f(b) then f(x) is left continuous at x = a.

a) False

b) True

View Answer

Explanation: A function is said to be continuous when it is both left continuous and right continuous. Mathematical expression for a function f is left continuous on (a,b) is lim

_{x→a-}f(x)=f(a).

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