Prove that the function f defined by

remains discontinuous at x=0, regardless the choice of k.

Given,

…(1)

We need to prove that f(x) is discontinuous at x = 0 irrespective of the value of k.

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)

Now, We need to prove that f(x) is discontinuous at x = 0 irrespective of the value of k

If we show that,

Then there will not be involvement of k in the equation & we can easily prove it.

So let’s take LHL first –

LHL =

⇒ LHL =

⇒ LHL =

∵ h > 0 as defined in theory above.

∴ |-h| = h

∴ LHL =

⇒ LHL =

∴ LHL = …(2)

Now Let’s find RHL,

RHL =

⇒ RHL =

⇒ RHL =

∵ h > 0 as defined in theory above.

∴ |h| = h

∴ RHL =

⇒ RHL =

∴ RHL = …(3)

Clearly form equation 2 and 3,we get

LHL ≠ RHL

Hence,

f(x) is discontinuous at x = 0 irrespective of the value of k.