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Mathematics Question on Differentiability

The function f(x)=∣x−10∣f (x) = |x-10| , xx is real number, is

A

differentiable every where but not continuous at x=10x=10

B

continuous everywhere but not differentiable at x=10x=10

C

continuous everywhere and differentiable at all points

D

continuous every where but not differentiable at x=0x=0

Answer

continuous everywhere but not differentiable at x=10x=10

Explanation

Solution

f(x) = |x - 10| = \begin{cases} x-10 & \text{if x \ge 10} \\\[2ex] -(x-10) & \text{ifx < 10 } \end{cases}
Continuity at x=10:x = 10:
L.H.S=lim⁡x→10−f(x)=lim⁡h→0f(10−h)L.H.S = \lim\limits_{x\to10^{-}} f\left(x\right) = \lim\limits _{h\to0} f\left(10-h\right)
=lim⁡h→0−(10−h−10)=0= \lim\limits_{h\to0}-\left(10-h -10\right) = 0
R.H.L=lim⁡x→10+f(x)=lim⁡h→0f(10+h)R.H.L = \lim\limits_{x\to10^{+}} f\left(x \right) =\lim\limits_{h \to0}f(10+h) =lim⁡h→0(10+h−10)=0=\lim\limits_{h\to0}\left(10+ h -10\right) =0
Also, f(10)=∣10−10∣=0f\left(10\right) = \left|10-10\right| = 0
Since, L.H.L.=R.H.L.=f(10)L.H.L. = R.H.L. = f\left(10\right)
∴f\therefore f is continuous at x=10 x= 10
⇒f\Rightarrow f is continuous everywhere on real numbers.
Differentiability at x=10 x = 10:
R.H.D.=lim⁡h→0f(10+h)−f(10)hR. H .D. = \lim\limits_{h\to0} \frac{f(10 +h)-f(10)}{h}
lim⁡h→010+h−10−0h=1\lim\limits_{h\to0} \frac{10 + h -10 -0}{h} = 1
L.H.D.=lim⁡h→0f(10−h)−f(10)−hL.H.D. =\lim\limits _{h\to0} \frac{f(10-h)-f(10)}{-h}
=lim⁡h→0−(10−h−10)−0−h=−1=\lim\limits_{h\to0} \frac{-(10-h -10)-0}{-h} =-1
Since, L.H.D≠R.H.D.L.H.D \ne R.H.D.
∴f\therefore f is not differentiable at x=10 x = 10.