# Which of the following is true ? Option 1) $f(x)= 1/|x|$   is continous at x= 0 Option 2) $f(x)= |x|/x$   is continous at x= 0 Option 3) $f(x)= [x]$   is continous at x= 1([.]=G.I.F) Option 4) $f(x)= [x]$   is continous at x= 1.5([.]= G.I.F)

H Himanshu

As we have learned

Continuity at a point -

A function f(x)  is said to be continuous at  x = a in its domain if

1.  f(a) is defined  : at  x = a.

$2. \lim_{x\rightarrow a}\:f(x)\:exists\:means\:limit\:x\rightarrow a$

of  f(x) at  x = a exists from left and right.

$3. \lim_{x\rightarrow a}\:f(x)=f(a)\:then\:the\:limit\:equals \:the\:value\:at\:x=a$

-

In(A) , f(x) is not defined at x=0

In (B) , f(x) is not defined at x=0 and also LHL$\neq$RHL

In (C) , f(x) is defined at  x= 1, f(1) =1

But f(1) LHL, RHL all are not same

In (D) f(1.5)= $f(1.5^{+})= f(1.5^{-})=0$

Option 1)

$f(x)= 1/|x|$   is continous at x= 0

Option 2)

$f(x)= |x|/x$   is continous at x= 0

Option 3)

$f(x)= [x]$   is continous at x= 1([.]=G.I.F)

Option 4)

$f(x)= [x]$   is continous at x= 1.5([.]= G.I.F)

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