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.


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