# CONTINUITY OF A ABSOLUTE VALUE FUNCTION AT A POINT

For a function to be continuous at a point,

• It must be defined at that point
• Its limit must exist at the point
• The value of the function at that point must be equal to the value of the limit at that point.

If the function f(x) is continuous at a point x = a, then

Problem 1 :

Check if the function is continuous at x = 0

f(x) = 2x - |x|

Solution :

Step 1 :

Defining the given absolute value function as piece wise function.

Step 2 :

Evaluating the left hand limit :

Evaluating the right hand limit :

Both left hand limit and right hand limit they are equal.

So, the function is continuous at x = 0.

Problem 2 :

Discuss the continuity of the function of given by

f(x) = |x - 1| + |x - 2| at x = 1 and x = 2.

Solution :

The condition for three pieces will be

x < 1, 1 ⩽ x < 2 and x ⩾ 2

• Choosing one of the value in the first condition (x < 1) and applying the absolute value function, we get negative for x - 1 and for x - 2.
• Choosing one of the value in the second condition (1 ⩽ x < 2) and applying the absolute value function, we get positive for x - 1 and negative for x - 2.
• Choosing one of the value in the third condition (x ⩾ 2) and applying the absolute value function, we get positive for x - 1 and x - 2.

Continuity at x = 1 :

Evaluating the left hand limit :

Evaluating the right hand limit :

So, the function is continuous at x = 1.

Continuity at x = 2 :

So, the function is continuous at x = 2.

Problem 3 :

Show that

is discontinuous at x = 0.

Solution :

Checking the left hand and right hand limit :

f(0) = 2

So, the function f(x) is discontinuous at x = 0.

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