Given,

…(1)

We need to check its continuity at x = 0

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 according to above theory-

f(x) is continuous at x = 4 if -

Clearly,

LHL = {using equation 1}

∵ h > 0 as defined above.

∴ |-h| = h

⇒ LHL =

As cos (1/h) is going to be some finite value from -1 to 1 as h→0

∴ LHL = 0 × (finite value) = 0 …(2)

Similarly we proceed for RHL-

RHL = {using equation 1}

∵ h > 0 as defined above.

∴ |h| = h

⇒ RHL =

As cos (1/h) is going to be some finite value from -1 to 1 as h→0

∴ RHL = 0 × (finite value) = 0 …(3)

And,

f(0) = 0 {using eqn 1} …(4)

Clearly from equation 2 , 3 and 4 we can say that

∴ f(x) is continuous at x = 0