Discuss the continuity of the function f, where f is defined by`f(x)={3, if0lt=xlt=1 4, if1

Explanation

`f(x){{:(3"," if 0 le x le 1),(4"," if 1lt x lt 3 ),(5"," if 3 le x le 10):}`
Here we will discuss the continuity of f(x) at x =1 and x = 3 only.
at x 1
f(1) = 3
R.H.L. = ` underset (x to 1^(+))(lim)f(x)`
Let 3 - h = x
`rArr 1 + h to 1`
`rArr h to 0`
`=underset (h to 0)(lim) f(1+h)`
`=underset (h to 0)(lim) 4 = 4`
`:'` f(1) `ne` R.H.L.
`:'` f(x) is discontinuous at x = 1
at x = 3
f(3) = 5
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(1+h)`
`=underset (h to 0)(lim) 4 = 4`
`:'` f(3) `ne` L.H.L.
`:.` f(x) is discontinuous at x = 3 .
f(x) is a constant function at all remaining points.
`:.` f(x) is continious at all points except x =1 and x =3.