To determine the continuity and differentiability of the function
f(x) at
x=1, let's first analyze its continuity.
We are given the piecewise function:
f(x)={| x | ,‌‌‌0≤x≤1 |
| 2x−1 | ,‌‌‌x>1 |
. To check for continuity at
x=1, we need to ensure that the left-hand limit ( LHL
), right-hand limit (RHL), and the function value at
x=1 are all equal. Let's evaluate these:
- Left-hand limit at
x=1 :
f(x)=x=1- Right-hand limit at
x=1 :
f(x)=(2x−1)=2(1)−1=1- Function value at
x=1 :
f(1)=1 Since LHL, RHL, and the function value at
x=1 are all equal, the function
f(x) is continuous at
x=1.
Next, let's check for differentiability at
x=1. We need to verify if the left-hand derivative (LHD) and righthand derivative (RHD) are equal at
x=1 :
- Left-hand derivative at
x=1 :
- Right-hand derivative at
x=1 :
Since the LHD is 1 and the RHD is 2 , the derivatives are not equal. Therefore, the function
f(x) is not differentiable at
x=1.
Hence, the correct option is:
Option C
f is continuous but not differentiable at
x=1