Discuss the continuity of the function f, where f is defined by f(x) ={-2,if x≤ -1, 2x if -1 < x ≤ 1,2,if x>1

Question

Discuss the continuity of the function f, where f is defined by

f(x) =-2,if x≤ -1, 

f(x) = 2x, if -1 < x ≤ 1,

f(x) = 2,if x>1

Answer 1

The given function is defined at all points of the real line. Let c be a point on the real line.
Case I:

Therefore, f is continuous at all points x, such that x < −1 

Case II:

Therefore, f is continuous at x = −1
Case III:

Therefore, f is continuous at all points of the interval (−1, 1). 

Case IV:

Therefore, f is continuous at x = 2 

Case V:

Therefore, f is continuous at all points x, such that x > 1 

Thus, from the above observations, it can be concluded that f is continuous at all points of the real line.