Discuss the continuity of the function on the closed interval.

Question

Discuss the continuity of the function on the closed interval.

Function	Interval
f(x) =	2 − x,	x ≤ 0	2 + 1
2	x,	x > 0
[−1, 2]

The function is continuous because lim x→0⁻ f(x) = lim x→0⁺ f(x) = f(0) = 2.

The function is discontinuous because all piecewise functions are discontinuous at their domain boundaries.    The function is continuous because lim x→−1⁺ f(x) and lim x→2⁻ f(x) both exist.

The function is discontinuous because f(−1) ≠ f(2).

The function is continuous because the domain boundary is not inside the interval [−1, 2].

 

Answer 1

hi,

1)The function is continuous because lim x→0⁻ f(x) = lim x→0⁺ f(x) = f(0) = 2.

2)The function is discontinuous because all piecewise functions are discontinuous at their domain boundaries.    

3)The function is continuous because lim x→−1⁺ f(x) and lim x→2⁻ f(x) both exist.

4)The function is discontinuous because f(−1) ≠ f(2).

5)The function is continuous because the domain boundary is not inside the interval [−1, 2].

#1 is wrong because lim x→0⁻ =2 and f(x) = lim x→0⁺ =0 so this function is discontinuous

#3 is wrong because there is not right reason for being continues a function

#4 is wrong because there is not right reason for being continues a function at initial and end of boundaries the same value

#5 is wrong because in boundaries [-1,2] exist x=0 that function is discontinuous

Be careful, as a result you cam make a decision #2 is a correct choice BUT I dont agree because all piecewise functions are not discontinuous at their domain, we can make an example that piecewise function is continues and has boundaries.

Hope was useful,

Minoo