Show that the function `g(x)=x-|x|` is discontinuous at all integral points. Here `[x]` denotes the greatest integer function.

Explanation

g(x) = x [x]
Let a be any integer.
`:.` [a + h] = a, [a] = a
and [a- h ] = a - 1
`:.` g (a) = a - [a]
= a - a = 0
L.H.L. = ` underset (x to a^(-))(lim)g(x)`
Let a - h = x
`rArr a - h to a `
`rArr h to 0`
` underset (h to 0)(lim)g(a - h)`
` underset (h to 0)(lim) a - h-[a - h]`
` underset (h to 0)(lim) a - h-(a - 1)=1 `
`:'` g(a) `ne` L.H.L.
`:.` g(x) is discontinuous for each integral value
Hence Proved.