Prove that the greatest integer function defined by f (x) = [x], 0 < x < 3 is not differentiable at x = 1 and x = 2.
Answers
we have to prove that the greatest integer function defined by f(x) = [x], 0 < x < 3 is not differentiable at x = 1 and 2.
proof : any function, y = f(x) is differentiable at x = a, where a lies in the domain of f(x), only when
Left hand differentiation (LHD) = Right hand differentiation (RHD)
at x = 1
LHD = lim(h →0) {f(x) - f(x - h)}/h
= lim(h→0) {f(1) - f(1 - h)}/h
= lim(h →0) {[1] - [1 - h]/h
= (1 - 0)/h
= 1/0 = not defined
RHD = lim(h→0) {f(x + h) - f(x)}/h
= lim(h→0) {f(1 + h) - f(1)}/h
= lim(h→0) {[1 + h] - [1]}/h
= (1 - 1)/h
= 0
here LHD ≠ RHD
Therefore,f(x) is not differentiable at x = 1
at x = 2
LHD = lim(h →0) {f(x) - f(x - h)}/h
= lim(h→0) {f(2) - f(2 - h)}/h
= lim(h →0) {[2] - [2 - h]/h
= (2 - 1)/h
= 1/0 = not defined
RHD = lim(h→0) {f(x + h) - f(x)}/h
= lim(h→0) {f(2 + h) - f(2)}/h
= lim(h→0) {[2 + h] - [2]}/h
= (2 - 2)/h
= 0
Therefore, f(x) is not differentiable at x = 2