# Prove that the function f defined byremains 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 (h0)

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.

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