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.e., k < 1, or k = 1 or k >1

Now, Case I: k < 1

Then, f(k) = k² + 1

= k² + 1= f(k)

Thus,

Hence, f is continuous at all real number less than 1.

Case II: k = 1

Then, f(k) = f(1) = 1 + 1 = 2

= 1² + 1 = 2

= 1 + 1 = 2

Hence, f is continuous at x = 1.

Case III: k > 1

Then, f(k) = k + 1

= k + 1 = f(k)

Thus,

Hence, f is continuous at all real number greater than 1.