Find all the points of discontinuity of the function f defined by`f(x)={x+2, ifx<1 0ifx=1x-2ifx >1`

Explanation

Case I : When `x = 1`


`f(x)` is continuous at `x = 1`

If L.H.L = R.H.L = `f(1)`


If `lim_(xrarr1^-) f(x) = lim_(xrarr1^+) f(x) = f(1)`

If `lim_(xrarr1^-) f(x) = lim_(xrarr1^+) f(x) = f(1)`

LHL at ` x rarr 1`

`lim_(xrarr1^-)f(x) = lim_(hrarr0) f(1-h)`

= `lim_(hrarr0)(1-h) + 2`

= `lim_(hrarr0) (3-h)`

= `3 - 0`

= `3`


RHL at `x rarr 1`

`lim_(xrarr1^+)f(x)` = `lim_(hrarr0)(1+h) - 2`

= `lim_(hrarr0)(-1+h) `

`= -1 + 0`

`= -1`



Since L.H.L `!=` R.H.L `f(x)` is not continuous at `x = 1`

Case II : When `x < 1` ,

`f(x)` = `x + 2`

Since this is a polynomial

It is continuous

`:. f(x)` is continuous for `x < 1`

Case III : When `x > 1`


For `x> 1`

` f(x) = x-2`

Since this is a polynomial

It is continuous

`:. f(x)` is continuous for `x > 1`

Hence , only `x = 1` is the point is discontinuity

`f` is continuous at all real numbers except 1

Thus , `f` is continuous for `x in R - {1}`