consider the following statements 1 the function f x

Consider the following statements :
1. The function $$f\left( x \right) = $$ greatest integer $$ \leqslant x,\,x\, \in \,R$$ is a continuous function.
2. All trigonometric function are continuous on $$R.$$
Which of the statements given above is/are correct ?

A. 1 only

B. 2 only

C. Both 1 and 2

D. Neither 1 nor 2

Answer : Neither 1 nor 2

Solution :
Here, greatest integer function $$\left[ x \right]$$ is discontinuous at its integral value of $$x,\,\cot \,x$$ and $${\text{cosec}}\,x\,$$ are discontinuous at $$0,\,\pi ,\,2\pi $$ etc. and $$\tan \,x$$ and $$sec \,x$$ are discontinuous at $$x = \frac{\pi }{2},\,\frac{{3\pi }}{2},\,\frac{{5\pi }}{2}$$ etc. Therefore the greatest integer function and all trigonometric function are not continuous for $$x\, \in \,R$$
Therefore, neither (1) nor (2) are true.

Releted MCQ Question on
Calculus >> Continuity

Releted Question 1

For a real number $$y,$$ let $$\left[ y \right]$$ denotes the greatest integer less than or equal to $$y:$$ Then the function $$f\left( x \right) = \frac{{\tan \left( {\pi \left[ {x - \pi } \right]} \right)}}{{1 + {{\left[ x \right]}^2}}}$$ is-

A. discontinuous at some $$x$$

B. continuous at all $$x,$$ but the derivative $$f'\left( x \right)$$ does not exist for some $$x$$

C. $$f'\left( x \right)$$ exists for all $$x,$$ but the second derivative $$f'\left( x \right)$$ does not exist for some $$x$$

D. $$f'\left( x \right)$$ exists for all $$x$$

View Answer

Releted Question 2

The function $$f\left( x \right) = \frac{{\ln \left( {1 + ax} \right) - \ln \left( {1 - bx} \right)}}{x}$$ is not defined at $$x = 0.$$ The value which should be assigned to $$f$$ at $$x = 0,$$ so that it is continuous at $$x =0,$$ is-

A. $$a-b$$

B. $$a+b$$

C. $$\ln a - \ln b$$

D. none of these

View Answer

Releted Question 3

The function $$f\left( x \right) = \left[ x \right]\cos \left( {\frac{{2x - 1}}{2}} \right)\pi ,\,\left[ . \right]$$ denotes the greatest integer function, is discontinuous at-

A. all $$x$$

B. All integer points

C. No $$x$$

D. $$x$$ which is not an integer

View Answer

Releted Question 4

The function $$f\left( x \right) = {\left[ x \right]^2} - \left[ {{x^2}} \right]$$ (where $$\left[ y \right]$$ is the greatest integer less than or equal to $$y$$ ), is discontinuous at-

A. all integers

B. all integers except 0 and 1

C. all integers except 0

D. all integers except 1

View Answer

Consider the following statements :
1. The function $$f\left( x \right) = $$ greatest integer $$ \leqslant x,\,x\, \in \,R$$ is a continuous function.
2. All trigonometric function are continuous on $$R.$$
Which of the statements given above is/are correct ?

Releted MCQ Question on
Calculus >> Continuity

For a real number $$y,$$ let $$\left[ y \right]$$ denotes the greatest integer less than or equal to $$y:$$ Then the function $$f\left( x \right) = \frac{{\tan \left( {\pi \left[ {x - \pi } \right]} \right)}}{{1 + {{\left[ x \right]}^2}}}$$ is-

The function $$f\left( x \right) = \frac{{\ln \left( {1 + ax} \right) - \ln \left( {1 - bx} \right)}}{x}$$ is not defined at $$x = 0.$$ The value which should be assigned to $$f$$ at $$x = 0,$$ so that it is continuous at $$x =0,$$ is-

The function $$f\left( x \right) = \left[ x \right]\cos \left( {\frac{{2x - 1}}{2}} \right)\pi ,\,\left[ . \right]$$ denotes the greatest integer function, is discontinuous at-

The function $$f\left( x \right) = {\left[ x \right]^2} - \left[ {{x^2}} \right]$$ (where $$\left[ y \right]$$ is the greatest integer less than or equal to $$y$$ ), is discontinuous at-

Practice More Releted MCQ Question on
Continuity

Practice More MCQ Question on Maths Section

Consider the following statements : 1. The function $$f\left( x \right) = $$ greatest integer $$ \leqslant x,\,x\, \in \,R$$ is a continuous function. 2. All trigonometric function are continuous on $$R.$$ Which of the statements given above is/are correct ?

Releted MCQ Question on Calculus >> Continuity

For a real number $$y,$$ let $$\left[ y \right]$$ denotes the greatest integer less than or equal to $$y:$$ Then the function $$f\left( x \right) = \frac{{\tan \left( {\pi \left[ {x - \pi } \right]} \right)}}{{1 + {{\left[ x \right]}^2}}}$$ is-

The function $$f\left( x \right) = \frac{{\ln \left( {1 + ax} \right) - \ln \left( {1 - bx} \right)}}{x}$$ is not defined at $$x = 0.$$ The value which should be assigned to $$f$$ at $$x = 0,$$ so that it is continuous at $$x =0,$$ is-

The function $$f\left( x \right) = \left[ x \right]\cos \left( {\frac{{2x - 1}}{2}} \right)\pi ,\,\left[ . \right]$$ denotes the greatest integer function, is discontinuous at-

The function $$f\left( x \right) = {\left[ x \right]^2} - \left[ {{x^2}} \right]$$ (where $$\left[ y \right]$$ is the greatest integer less than or equal to $$y$$ ), is discontinuous at-

Practice More Releted MCQ Question on Continuity

Practice More MCQ Question on Maths Section

Consider the following statements :
1. The function $$f\left( x \right) = $$ greatest integer $$ \leqslant x,\,x\, \in \,R$$ is a continuous function.
2. All trigonometric function are continuous on $$R.$$
Which of the statements given above is/are correct ?

Releted MCQ Question on
Calculus >> Continuity

Practice More Releted MCQ Question on
Continuity