Examine the continuity of the function `f(x)=`
`{{:(3x-2,xlt=0),(x+1,x >0):}`
a t `x=0`
Also sketch the graph of this function.

Explanation

Given function `f(x)={{:(3x-2,xlt=0),(x+1,x >0):}`
a t `x=0`
LHL at `x=0`,
`lim_(x->0^-)f(x)`
`=lim_(h->0)f(0-h)`
`=lim_(h->0)f(-h)`
`=lim_(h->0)3(-h)-2`
`=lim_(h->0)-3h-2`
`=-2`
RHL at `x=0`,
`lim_(x->0^+)f(x)`
`=lim_(h->0)f(0+h)`
`=lim_(h->0)f(h)`
`=lim_(h->0)h+1`
`=1`
`LHL ne RHL`
therefore this function is discontinuous at `x=0`