Question

Let f(x)=[x], then f(x) is : *

1 point

differentiable for all x∈R-I

differentiable for all x∈R

continuous for all x∈R

continuous nowhere

​

Answer 1

Given : f(x)=[x]

To find : Choose correct statement

differentiable for all x∈R-I

differentiable for all x∈R

continuous for all x∈R

continuous nowhere

Solution:

f(x) = [x}

if x is not integer

I < x < I + 1    I -integer

Then f(x - h) = f(x) = f(x+h)  = I  hence continues

x = I

then f(x-h) = I - 1

f(x) = f(x+h)  = I

hence not continuous at  Integer

Hence  continues for all x ∈ R - I

I < x < I + 1    I -integer

f(x - h) = f(x) = f(x+h)  = I

LHD = h ->0   ( f(x) - f(x-h) )/h     = 0/h = 0

RHD = h ->0  (f(x+h) - f(x)) /h     = 0/h = 0

LHD=RHD

Differentiable at x ∈ R - I  

x = I  

f(x-h) = I - 1

f(x) = f(x+h)  = I

LHD = h ->0   ( f(x) - f(x-h) )/h     = 1/h

RHD = h ->0  (f(x+h) - f(x)) /h     = 0/h

LHD  ≠ RHD

Not differentiable at Integers

f(x)=[x]  differentiable for all x∈R-I

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