Find all points of discontinuity of f, where f is defined by`f(x)={|x|+3, ifxlt=-3-2x , if-3

Explanation

`f(x)={{:(|x|+3, if x le -3),(-2x, if -3 lt x lt 3 ),(6x + 2, if x ge3):}`
Here we will discuss the continuity of the function
at x = - 3 and x - 3.
at x = -3
Let -3 + h = x
`rArr 3 + h to -3`
`rArr h to 0 `
f(-3) = |-3| + 3 = 3 + 3 = 6
R.H.L. = ` underset (x to 3^(+))(lim)f(x)`
` =underset (h to 0)(lim)f(-3 + h)`
`= underset (h to 0)(lim)f-2(-3 + h)`
= 2(-3 + 0) = 6
L.H.L. = ` underset (x to 3^(-))(lim)f(x)`
Let -3 + h = x
`rArr -3 h to -3`
` rArr h to 0`
` =underset (h to 0)(lim)f(-3 - h)` ltbrge ` underset (h to 0)(lim)|-3-h|+3`
`|-3-0|+= 3 + 3=6`
`:'` L.H.L. = f(-3) = R.H.L.
`:.` f(x) is continuous at x = - 3
at x = 3
Let 3 - h = x
`rArr 3 - h to 3`
`rArr h to 0 `
f(3)` = 6xx3 + 2 = 20`
L.H.L. = ` underset (x to 3^(-))(lim)f(x)`
=` underset (h to 0)(lim)f(3-h)`
`=underset (h to 0)(lim) -2(3-h)`
` -2(3 -0) = -6`
`:'` f(3) `ne` L.H.L.
`:.` f(x) is discontinous at x = 3