` f(x){{:(x^(3) - 3"," if x le 2 ),(x^(2) + 1"," if x gt 2 ):}`

Explanation

` f(x){{:(x^(3) - 3"," if x le 2 ),(x^(2) + 1"," if x gt 2 ):}`
If `x lt 2 then f(x) = x^(3) - 3 `is a polynomial function.
If `x gt then f(x) = x^(2) + 1` is a polynomial function.
A polynomial function is always continuous.
Now, at x = 2
`f(2) 2^(3)- 3 = 8 - 3 = 5`
L.H.L. = ` underset (x to 2^(-))(lim)f(x)`
Let 2 - h = x
`rArr 2 - h to 2 `
`rArr h to 0`
`=underset (h to 0)(lim) f(2-h)`
`=underset (h to 0)(lim) (2 - h)^(3) -3`
`= (2 - 0)^(3) - 3 = 8 - 3 = 5`
R.H.L. = ` underset (x to 2^(+))(lim)f(x)`
Let 2 + h = x
`rArr 2 + h to 2`
`rArr h to 0`
`=underset (h to 0)(lim) f(2+h)`
`=underset (h to 0)(lim) (2 - h)^(2) + 1 `
`=(2 + 0)^(2) + 1 = 4 + 1 = 5`
`:'` L.H.L. = f(2) = R.H.L.
`:.` f(x), is continuous at x = 2
Therefore , f(x) is always continuous.