Continuity

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Question 1

Which one of the following functions is continuous at x = 3?

  • [Tex]f(x)=\left\{\begin{array}{lll}2, & \text { if } & x=3 \\ x-1, & \text { if } & x>3 \\ \frac{x+3}{3}, & \text { if } & x<3\end{array}\right.[/Tex]

  • [Tex]f(x)=\left\{\begin{array}{lll}4, & \text { if } & x=3 \\ 8-x & \text { if } & x \neq 3\end{array}\right.[/Tex]

  • [Tex]f(x)=\left\{\begin{array}{lll}x+3, & \text { if } & x \leq 3 \\ x-4, & \text { if } & x>3\end{array}\right.[/Tex]

  • [Tex]f(x)=\frac{1}{x^3-27}, [/Tex]if x ≠ 3

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Question 2

Consider the following two statements about the function f(x) = |x|

P. f(x) is continuous for all real values of x
Q. f(x) is differentiable for all real values of x

Which of the following is TRUE?

  • P is true and Q is false.

  • P is false and Qis true.

  • Both P and Q are true

  • Both P and Q are false.

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Question 3

Consider the function y = |x| in the interval [-1,1]. In this interval, the function is

  • continuous and differentiable

  • continuous but not differentiable

  • differentiable but not continuous

  • neither continuous nor differentiable

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Question 4

Let f be a function defined by

[Tex]f(x) =\begin{cases} x^2 & \text{for } x \leq 1 \\ax^2 + bx + c & \text{for } 1 < x \leq 2 \\x + d & \text{for } x > 2\end{cases}[/Tex]

Find the values for the constants a, b, c and d so that f is continuous and differentiable everywhere on the real line.

  • a = -0.5, b = 3, c = -1.5, d = 0.5

  • a = -1, b = 1, c = 1, d = 1

  • a = -2, b = 1, c = -1.5, d = 0.5

  • None of These

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Question 5

A function f(x) is continuous in the interval [0, 2]. It is known that f(0) = f(2) = -1 and f(1) = 1. Which one of the following statements must be true?

  • There exists a y in the interval (0, 1) such that f(y) = f(y+1)

  • For every y in the interval (0, 1), f(y) = f(2-y)

  • The maximum value of the function in the interval (0, 2) is 1

  • There exists a y in the interval (0, 1) such that f(y) = -f(2-y)

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Question 6

Which one of the following functions is continuous at x = 3?

  •  
  •  
  •  
  •  
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Question 7

The function y = |2 –3x|

  • is continuous ∀ x ∈ R and differentiable ∀ x ∈ R

  • is continuous ∀ x ∈ R and differentiable ∀ x ∈ R except at x = 3/2

  • is continuous ∀ x ∈ R and differentiable ∀ x ∈ R except at 2/3

  • is continuous ∀ x ∈ R except x = 3 and differentiable ∀ x ∈ R

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Question 8

Consider the function f(x) = |x3|, where x is real. Then the function f(x) at x = 0 is

  • Continuous but not differentiable

  • Once differentiable but not twice

  • Twice differentiable but not thrice

  • Three differentiable

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Question 9

The value of x for which the function

 

is not continuous are

  • 4 and –1

  • 4 and 1

  • –4 and 1

  • –4 and –1

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Question 10

The function f(x) = |x + 1| on the interval [–2, 0] is

  • Continuous and differentiable

  • Continuous on the integers but not differentiable at all points

  • Neither continuous nor differentiable

  • Differentiable but not continuous

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