Final answer:
A piecewise function is a function that is defined by different expressions on different intervals. The continuity of a piecewise function is determined by whether the separate expressions are continuous at their respective intervals.
Step-by-step explanation:
A piecewise function is a function that is defined by different expressions on different intervals. The continuity of a piecewise function is determined by whether the separate expressions are continuous at their respective intervals. Let's analyze each function:
a) x²+1: This function is continuous on its entire domain because it is a polynomial function, and polynomial functions are continuous everywhere.
b) √x: This function is continuous on its interval of definition, which is x ≥ 0. The square root function is continuous for non-negative values of x.
c) 1/x: This function is continuous for all x ≠ 0. The reciprocal function is continuous everywhere except at x = 0, where it has a vertical asymptote.
d) |x|: This function is continuous except at x = 0, where it has a sharp turn or corner. The absolute value function has a kink at x = 0.