Find which of the functions is continuous or discontinuous at the indicated points:

Question

Philoid · Accepted Answer

Given,

…(1)

We need to check its continuity at x = 1

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 = 1 if -

Clearly,

LHL = {using equation 1}

∴ LHL = (1-0)²/2 = 1/2 …(2)

Similarly, we proceed for RHL-

RHL = {using eqn 1}

⇒ RHL =

∴ RHL = 2(0)² + 0 + 1/2 = 1/2 …(3)

And,

f(1) = 1²/2 = 1/2 {using eqn 1} …(4)

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

∴ f(x) is continuous at x = 1