Q. 234.0( 2 Votes )

Show that f(x) = |x – 5| is continuous but not differentiable at x = 5.

Answer :

Given,


f(x) = |x - 5| …(1)


We need to prove that f(x) is continuous but not differentiable at x = 5


A function f(x) is said to be continuous at x = c if,


Left hand limit(LHL at x = c) = Right hand limit(RHL at x = c) = f(c).


Mathematically we can represent it as-



Where h is a very small number very close to 0 (h0)


And a function is said to be differentiable at x = c if it is continuous there and


Left hand derivative(LHD at x = c) = Right hand derivative(RHD at x = c) = f(c).


Mathematically we can represent it as-




Finally we can state that for a function to be differentiable at x = c



Checking for the continuity:


Now according to above theory-


f(x) is continuous at x = 5 if -



LHL =


LHL = {using equation 1}


LHL =


LHL = |-0| = 0


LHL = 0 …(2)


Similarly,


RHL =


RHL = {using equation 1}


RHL =


RHL = |0| = 0 …(3)


And, f(5) = |5-5| = 0 {using equation 1} …(4)


From equation 2,3 and 4 we observe that:



f(x) is continuous at x = 5. So we will proceed now to check the differentiability.


Checking for the differentiability:


Now according to above theory-


f(x) is differentiable at x = 2 if -



LHD =


LHD = {using equation 1}


As h > 0 as defined in theory above.


|-h| = h


LHD =


LHD = -1 …(5)


Now,


RHD =


RHD = {using equation 1}


As h > 0 as defined in theory above.


|h| = h


RHD =


RHD = 1 …(6)


Clearly from equation 5 and 6,we can conclude that-


(LHD at x=5) ≠ (RHD at x = 5)


f(x) is not differentiable at x = 5 but continuous at x = 5.


Hence proved.


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