Find all points of discontinuity of f, where f is defined by`f(x)={{:(x^3-3, ifxlt=2),(x^2+1, ifx<2):}`

Explanation

The given function f is `f(x)={{:(x^3-3, ifxlt=2),(x^2+1, ifx<2):}`


The given function f is defined at all points of the real line.
Let c be a point on the real line .
Case I :

If `c<2` , then f (c) = `c^3` - 3 and `lim_(xrarrc)`f(x) = `lim_(xrarrc) (x^3 -3 ) = c^3 -3`

`:. lim_(xrarrc) f(x) = f(c)`

Therefore , f is continuous at all points x, such that x < 2




Case II:
If c =2 , then `f (c) = f(2) = 2^3-3 = 5`

`lim_(xrarr2-) = lim_(xrarr2-) (x^3 - 3) = 2^3 - 3 = 5`

`lim_(xrarr2^-) f(x) = lim_(xrarr2^-) (x^2+1) = 2^2+1 =5`

`:. lim_(xrarr2^-) f(x) = f(2)` Therefore , f is continuous at x = 2
Case III :


`if e > 2 , then f(c) = c^2 + 1`

`:.lim_(xrarrc) (x)=f(c)`
Therefore , f is continuous at all points x , such that x > 2
Thus , the given function f is continuous at every point on the real line ,
Hence f has no point of discontinuity.