Which function is continuous across its domain?

a function is continuous, if the functional values are the same at the "connection" points of the various segments (and the segments themselves are continuous).

"continuous" simply means that the graph of the function is a continuous line without any "rips". it can have corners and such, but no "interruptions".

specifically it means : for every possible y value in the defined range of the function there is an x value that causes this y.

all defined segments are continuous functions.

so, let's look at

A. the first connection point is x=-2.

-2 + 6 = 0.5×(-2)²

4 = 0.5×4 = 2

4 = 2 is wrong. => here, at this point, the function "rips" apart and is not continuous.

B. x=-2

-2 + 4 = 0.5×(-2)²

2 = 2 is correct. continuous at this point.

second connection point x=4

0.5×4² = 20 - 3×4

0.5×16 = 20 - 12

8 = 8 is correct. continuous at this point

C. x=-2

-2 - 2 = 0.5×(-2)²

-4 = 2 is wrong. not continuous

D. x=-2

-2 + 4 = 0.5×(-2)²

2 = 2 is correct. continuous here.

now for x=4

4 + 4 = 25 - 3×4

8 = 25 - 12 = 13

8 = 13 is wrong. not continuous.