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 < 2, or k = 2 or k > 2

Now, Case I: k < 2

Then, f(k) = k³ - 3

= k³ - 3= f(k)

Thus,

Hence, f is continuous at all real number less than 2.

Case II: k = 2

Then, f(k) = f(2) = 2³ - 3 = 5

= 2³ - 3 = 5

= 2² + 1 = 5

Hence, f is continuous at x = 2.

Case III: k > 2

Then, f(k) = 2² + 1 = 5

= 2² + 1 = 5 = f(k)

Thus,

Hence, f is continuous at all real number greater than 2.