Question

Find all the points of discontinuity of f defined by f(x)= |x|-|x+1|

Answer 1

The given function is f (x) = |x| -|x +1

The two functions, g and h, are defined as

Clearly, g is defined for all real numbers.
Let c be a real number.
Case I:

Therefore, g is continuous at all points x, such that x < 0
Case II:

If c> 0 then g(c) = c

and \(\lim\limits_{x \to c} g(x) = \lim\limits_{x \to c} x = c\)

\(\lim\limits_{x \to c} g(x) = g(c)\)

Therefore, g is continuous at all points x, such that x > 0
Case III:

Therefore, g is continuous at x = 0
From the above three observations, it can be concluded that g is continuous at all points

Therefore, h is continuous at all points x, such that x < −1
Case II:

Therefore, h is continuous at x = −1
From the above three observations, it can be concluded that h is continuous at all points of the real line. g and h are continuous functions. Therefore, f = g − h is also a continuous function. Therefore, f has no point of discontinuity.

Answer 2

f(x) = |x| - |x + 1|

Let g(x) = |x|

& h(x) = |x + 1|

Since, we know that modules function is always continuous .

Therefore, |x| and |x + 1| is continuous everywhere.

Hence, g(x) & h(x) are continuous everywhere.

\(\therefore\) g(x) - h(x) is also continuous everywhere.

⇒ f(x) is continuous everywhere (\(\because\) f(x) = g(x) - h(x))

⇒ There is no point where function f(x) is dis continuous.