Given,

f(x) = |x| + |x – 1| …(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 eqn 1}

⇒ LHL =

∵ h > 0 as defined above and h→0

∴ |-h| = h

And (1 – h) > 0

∴ |1 – h| = 1 - h

⇒ LHL =

∴ LHL = 1 …(2)

Similarly we proceed for RHL-

RHL = {using eqn 1}

⇒ RHL =

∵ h > 0 as defined above and h→0

∴ |h| = h

And (1 + h) > 0

∴ |1 + h| = 1 + h

⇒ RHL =

∴ RHL = 1 + 2(0) = 1 …(3)

And,

f(1) = |1|+|1-1| = 1 {using eqn 1} …(4)

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

∴ f(x) is continuous at x = 1