Question

Test the continuity of the function f(x) = | x | + | x – 1 | in interval [- 1, 2].

Answer 1

Given, f(x) = | x | + | x – 1 |

It can be written as

Here, we will test of continuity at x = 0 and x = 1 only. 

Test of continuity at x = 0

Here, f(0) = 1

Left hand limit

f(0 – 0) = lim_h→0 f(0 – h)

= lim_h→0 1 – 2(0 – h)

= 1

Right hand limit

f(0 + 0) = lim_h→0 f(0 + h)

= 1

∴ f(0) = f (0 – c) = f (0 + 0)

So, at x = 0,f(x) is continuous

Now, at x = 1 test of continuity

Here, f(1) = 2 × 1 – 1 = 1

Left hand limit

f (1 – 0) = lim_h→0 f(1 – h)

= lim_h→0 1

Right hand limit

f (1 + 0) = lim_h→0 f(1 + h)

= lim_h→0 2(1 + h) – 1

= 2(1 + 0) – 1

= 2 – 1 = 1

∴ f (1) = f(1 – 0) = f(1 + 0)

So, at x = 1, f(x) is continuous.

∴ Function is continuous at x = 0 and x = 1

Hence, function is continuous in interval [-1,2].