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

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 (h→0)

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.