Select all the statements that accurately complete the sentence about the continuity of the following function.

Question

Select all the statements that accurately complete the sentence about the continuity of the following function.

ƒ(x) = {cos x if x < 0}

e^x. if x ≥ 0

The function ƒ.....?

1. is continuous on(-∞,∞)
2. is continuous on(-∞,∞), except at x = 0
3. has a removable discontinuity at x = 0
4. has a jump discontinuity at x = 0

Answer 1

Pauline,

1. The function cos(x) is continuous for (-∞,0) while the function e^x is continuous on [0,∞).

2. The only point that needed some more attention is x=0

The limit of f(x)=cos(x) when x->-0 is 1. The limit of f(x)=e^x when x>+0 is also 1. Therefore f(x) has a limit (equal to 1) at x=0 The value of f(0)=e⁰=1 . Therefore, the limit of f(x) when x->0 is equal to f(0), another words, f(x) is continuous at x=0.

Combined with the P.1, we conclude that the function f(x)

1. is continuous on(-∞,∞)

Hope it was helpful

Dr. Ariel B.

Answer 2

Since cos(0) = e⁰ = 1 there is no discontinuity, so

ƒ(x) is continuous on (–∞,∞)