It looks like the function is defined by
[tex]f(x) = \begin{cases} x^2 + 1 & \text{if } x < 1 \\ -x^3 + 2 & \text{if } x > 1 \end{cases}[/tex]
which we immediately is discontinuous at x = 1 because f(1) is not defined. One of the strict inequalities should probably have ≥ or ≤ involved.
In order for f(x) to be continuous at x = 1, we need have the one-sided limits agree:
[tex]\displaystyle \lim_{x\to1^-} f(x) = \lim_{x\to1} (x^2+1) = 2[/tex]
[tex]\displaystyle \lim_{x\to1^+} f(x) = \lim_{x\to1} (-x^3+2) = 1[/tex]
They do not agree, so f(x) is indeed discontinuous at x = 1, regardless of what value we pick for f(1). Graphically this corresponds to a jump discontinuity, where f(x) jumps from 2 down to 1 as x varies past x = 1.
[A and D]
