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  • Stuck here, help me understand: Which of the following function is not continous at all x being in the interval [1,3]?

Which of the following function is not continous at all x being in the interval [1,3]?

  • Option 1)

    f(x)= x^{2}

  • Option 2)

    f(x)= x^{3}

  • Option 3)

    f(x)= \sin x

  • Option 4)

    f(x)= [x]

 

Answers (1)

best_answer

As we have learned

Continuity from Right -

f(x) is said to be continuous in a closed interval [a, b] or

a\leq x\leq b    if

1.  f is continuous at each and every point in ( a, b)

2.  Right hand limit at  x = a  must exist and 

  \lim_{x\rightarrow a^{+}}\:f(x)=f(a)

3.  Left hand limit at  x = b must exist and

\lim_{x\rightarrow b^{-}}\:f(x)=f(b)

- wherein

 

 (A),(B),(C) are the function which are continous at every point in (1,3) and for coninuity at x=1 and x= 3 \lim_{x\rightarrow 1^{+}}f(x)= f(1)  and \lim_{x\rightarrow 3^{-}}f(x)= f(3) also holds true 

so (A),(B),(C ) are continous at every pointof [1,3]

 In (D), f(x) =[x] which will be discontinous at x=2 and x=3 both as \lim_{x\rightarrow 2^{+}}f(x),  \lim_{x\rightarrow 2^{-}}f(x) and f(2) are not all equal and \lim_{x\rightarrow 3^{-}}f(x)\neq f(3) 

\therefore discontinous at x= 2  and x= 3

 

 

 

 

 

 

 


Option 1)

f(x)= x^{2}

Option 2)

f(x)= x^{3}

Option 3)

f(x)= \sin x

Option 4)

f(x)= [x]

Posted by

Himanshu

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