निम्नलिखित फलन के सांतत्य और अवकलनीयता का x = 1 पर विचार कीजिए :
`f(x) = {{:(x + 2,"यदि",x le 1),(x-2,"यदि",x gt 1):}`

Explanation

x=1 पर सांतत्य -
`f(1) =1 +2 = 3 " " `…(1)
`LHL = underset (h to 0 ) lim f(1-h)`
` = underset (h to 0) lim (1-h+2) = 1+2+3" "`…(2)
`RHL = underset (x to 1+0) lim f(x)`
` = underset (h to 0)lim f(1+h)`
` = underset (h to 0 ) lim (1 + h - 2 )= 1- 2 = -1 " "` ...(3)
` :' LHL != RHL`
` :. f(x), x = 1 " पर संतत नहीं है ।"`
x=1 पर अवकलनीयता -
` Lf' (1) = underset (x to 0) lim (f(x)-f(1))/(x-1)`
` underset (h to 0 ) lim (f(1-h) - f(1))/(1-h-1)`
` underset (h to 0 ) lim ((1-h+2)-3)/-h`
` underset (h to 0 ) lim ((-h)/(-h))= underset (h to 0 )lim (1) =1" "`...(4)
` Rf'(1) = underset (x to 1^(+)) lim (f(x)- f (1))/(x-1)`

` = underset(h to 0 ) lim (f(1+h)-f(1))/(1+h-1)`
`underset (h to 0)lim ((1+h-2)-3)/-h `
` = underset ( h to 0) lim(h-4)/4`
` = underset (h to 0 ) lim (1-4/h)= - oo" "`...(5)
` :' Lf'(1) != Rf'(1)`
` :. f (x), x = 1 `पर अवकलनीय नहीं है ।