Final answer:
The function f(x) = x⁽ is continuous for all natural numbers m because polynomials are inherently continuous over the real numbers and the limit of f(x) as x approaches any point is equal to the function's value at that point.
Step-by-step explanation:
To prove that if m ∈ ℕ, then the function f(x) = x⁽ is continuous on ℝ, we must show that the function meets the criteria for continuity. A function is continuous at a point if, as x approaches the value from either direction, the value of the function approaches the value of the function at that point. For polynomial functions like f(x) = x⁽, this means that we must be able to draw the graph of the function without lifting our pen, which inherently includes the criteria that the function must be defined at the point and that the limit of the function as x approaches the value is equal to the function's value at that point.
For all natural numbers m, the function f(x) = x⁽ is indeed continuous since polynomials are continuous functions. This is because the limit of f(x) as x approaches any real number a is exactly f(a), and there are no discontinuities or undefined points for x in the real numbers.