Consider the following statements in respect of the function \(f(x)=[3x]\), where \([\cdot]\) is the greatest integer function: I. \(f(x)\) is continuous at \(x=1/3\). II. \(f(x)\) is differentiable at \(x=1/4\). Which of the statements given above is/are correct?
II only
At \(x=1/3\), \(3x=1\), and \([3x]\) jumps from \(0\) to \(1\) there, so \(f\) is discontinuous at \(x=1/3\) and statement I is false. At \(x=1/4\), \(3x=3/4\) is not an integer, so \([3x]=0\) is constant in a neighbourhood of \(x=1/4\), making \(f\) differentiable there with derivative \(0\); statement II is true.
Similar Questions
A function is defined as follows: \[ f(x) = \begin{cases} -\dfrac{x}{\sqrt{x^2}}, & x \ne 0, \\[6pt] 0, & x = 0. \end{cases} \] Which one of the following is correct in respect of the above function?
Let f : A → R, where A = R\(0) is such that \({\rm{f}}\left( {\rm{x}} \right) = \frac{{{\rm{x}} + \left| {\rm{x}} \right|}}{{\rm{x}}}\) . On which one of the following sets is f(x) continuous?
Consider the following statements for f(x) = e -|x| ;
1. The function is continuous at x = 0.
2. The function is differentiable at x = 0.
Which of the above statements is / are correct?
If the function \(\rm f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {a + bx,\;\;}&{x < 1}\\ {5,}&{x = 1}\\ {b - ax,}&{x > 1} \end{array}} \right.\) is continuous, then what is the value of (a + b)?
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Consider the following statements in respect of the function \(\rm f(x) = sin \left(\frac{1}{x^2}\right)\) , x ≠ 0:
1. It is continuous at x = 0, if f(0) = 0.
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Which of the above statements is/are correct?
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If \({\rm{f}}\left( {\rm{x}} \right) = \frac{{{{\rm{x}}^2} - 9}}{{{{\rm{x}}^2} - 2{\rm{x}} - 3}}\) , x ≠ 3 is continuous at x = 3, then which one of the following is correct?
Let f(x) be defined as follows: \({\rm{f}}\left( {\rm{x}} \right) = \left\{ {\begin{array}{*{20}{c}} {2{\rm{x}} + 1,{\rm{\;\;}} - 3 < {\rm{x}} < - 2}\\ {{\rm{x}} - 1,{\rm{\;\;}} - 2 \le {\rm{x}} < 0}\\ {{\rm{x}} + 2,{\rm{\;\;\;}}0 \le {\rm{x}} < 1} \end{array}} \right.\) Which one of the following statements is correct in respect of the above function?
\({\rm{f}}\left( {\rm{x}} \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{{{{\rm{e}}^{\rm{x}}} - 1}}{{\rm{x}}},{\rm{\;\;x}} > 0}\\ {0,{\rm{\;\;\;x}} = 0} \end{array}} \right.\)
Which one of the following statements is correct?
Important Questions from Continuity of a function
A function is defined as follows: \[ f(x) = \begin{cases} -\dfrac{x}{\sqrt{x^2}}, & x \ne 0, \\[6pt] 0, & x = 0. \end{cases} \] Which one of the following is correct in respect of the above function?
f(x) = x + |x| is continuous for
Let f : A → R, where A = R\(0) is such that \({\rm{f}}\left( {\rm{x}} \right) = \frac{{{\rm{x}} + \left| {\rm{x}} \right|}}{{\rm{x}}}\) . On which one of the following sets is f(x) continuous?
Consider the following statements for f(x) = e -|x| ;
1. The function is continuous at x = 0.
2. The function is differentiable at x = 0.
Which of the above statements is / are correct?
If the function \(\rm f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {a + bx,\;\;}&{x < 1}\\ {5,}&{x = 1}\\ {b - ax,}&{x > 1} \end{array}} \right.\) is continuous, then what is the value of (a + b)?
