फलन f(x) के सांतत्यता की जाँच `x = (1)/(2)` पर कीजिए जहाँ f(x) निम्नानुसार परिभाषित है -
`f(x) = {{:((1)/(2)+x",",x le x lt (1)/(2)),(1",",x=(1)/(2)),((3)/(2)+x",",(1)/(2) lt x le 1):}`

Explanation

(i) `f((1)/(2))=1`
(ii) R.H.L. `=underset(x rarr (1^(+))/(2))(lim)f(x)=underset(h rarr 0)(lim)f((1)/(2)+h)`
`=underset(h rarr 0)(lim)(3)/(2)+((1)/(2)+h)," "[because x = (1)/(2)+h gt (1)/(2)]`
`=underset(h rarr 0)(lim)(2+h)`
`=2 + 0 = 2`
(iii) L.H.L. `=underset(x rarr (1^(-))/(2))(lim)f(x)=underset(h rarr 0)(lim)f((1)/(2)-h)`,
`=underset(h rarr 0)(lim)(1)/(2)+((1)/(2)-h)" "[because x = (1)/(2)-h lt (1)/(2)]`
`=underset(h rarr 0)(lim)(1-h)`
`=1 - 0= 1`
`therefore" "underset(x rarr (1^(+))/(2))(lim)f(x)ne underset(x rarr (1^(+))/(2))(lim)f(x)=f((1)/(2))`
अत: फलन f(x) बिन्दु `x = (1)/(2)` पर संतत है ।