Examine the following functions for continuity.

f (x) = | x – 5|

The given function is

The function f is defined at all points of the real line.

Let k be the point on a real line.

Then, we have 3 cases i.ee, k < 5, or k = 5 or k >5

Now, Case I: k<5

Then, f(k) = 5 – k

= 5 – k = f(k)

Thus,

Hence, f is continuous at all real number less than 5.

Case II: k = 5

Then, f(k) = f(5) = 5 – 5 = 0

= 5 – 5 = 0

= 5 – 5 = 0

Hence, f is continuous at x = 5.

Case III: k > 5

Then, f(k) = k – 5

= k – 5 = f(k)

Thus,

Hence, f is continuous at all real number greater than 5.

Therefore, f is a continuous function.