Question

Suppose f:[−1,0]→R,g:[0,1]→R are continuous functions such that f(0)= g(0). Define h:[−1,1]→R via h(x)={f(x),g(x),​x≤0x>0​ Show that h is

Answer 1

To show that the function h(x) is continuous over the interval [-1,1], we need to prove that it is continuous at x=0, and also continuous on the left and right of x=0.

Step-by-step explanation:

To show that the function h(x) is continuous over the interval [-1,1], we need to prove that it is continuous at x=0, and also continuous on the left and right of x=0.

To show h(x) is continuous at x=0, we need to use the fact that f(0) = g(0) (given). Since f(x) is continuous over [-1,0], and g(x) is continuous over [0,1], we can write:

lim(h(x) as x approaches 0 from the left) = lim(f(x) as x approaches 0 from the left) = f(0) = g(0) (using the continuity of f(x))

lim(h(x) as x approaches 0 from the right) = lim(g(x) as x approaches 0 from the right) = g(0) (using the continuity of g(x))

Therefore, h(x) is continuous at x=0.

To show h(x) is continuous on the left of x=0, consider the function f(x). Since f(x) is continuous over [-1,0], h(x) is also continuous on the left of x=0.

To show h(x) is continuous on the right of x=0, consider the function g(x). Since g(x) is continuous over [0,1], h(x) is also continuous on the right of x=0.