Is the function continuous

Question

Is the function  continuous?

Question options:

1)	Yes
2)	No

Answer 1

The function is not continuous.

Explanation:

For a function to be continuous it needs to be continuous at each point. This is a piecewise function with two "branches" (one branch for x < 4 and another for x >= 4) and on each branch the function is continuous because 2x+1 and x^2+12 are continuous functions.

However, we still need to check if the function is continuous at x = 4. A function is continuous at p if f(p) is equal to the limit of f(x) as x approaches p. The limit from the left going towards 4 is 2(4)+1 = 9 while the limit from the right is (4)^2 + 12 = 16+12 = 28. (Here we are using the continuity of the branches to compute the limits by substitution) In this case, the left and right limits don't agree (9 != 28) so the limit at 4 is undefined. This means the function is not continuous at 4. In particular, the function is not continuous.

Quick Answer:

Note that the left hand limit is: lim x→4- f(x) = lim x→4- 2x+1 = [note 1] lim x→4 2x+1 = [note 2] 2(4)+1 = 8 + 1 = 9

Note that the right hand limit is: lim x→4+ f(x) = lim x→4+ x^2+12 = lim x→4 x^2+12 = 4^2+12 = 16 + 12 = 28

Since the left and right limits don't match, i.e. 9 != 28 we say the quantity lim x→4 f(x) does not exist.

Since the limit doesn't exist at 4, the function cannot be continuous at 4, so f(x) is not continuous.

Note 1: Here we are using the fact that 2x+1 is a continuous function so we can compute it's left hand limit by computing the usual limit.

Note 2: Here we are using the fact that 2x+1 is a continuous function. If g(x) is a continuous function then lim x→a g(x) = g(a)

Answer 2

No.

Solve each expression at x = 4, and you will see the values are different