Problem Analysis #common_04

Recorded Python call:

# The function f(x) is defined piecewise.\n# For continuity at x=1, left limit must equal right limit.\nleft_limit = 3*1 - 2\nright_limit = 1**2 - 3*1 + a\n# Solve left_limit = right_limit\nfrom sympy import symbols, Eq, solve\na = symbols('a')\neq = Eq(left_limit, right_limit.subs('a', a))\nsol = solve(eq, a)\nprint("Left limit:", left_limit)\nprint("Right limit expression:", right_limit)\nprint("Solution for a:", sol)

The function is continuous everywhere if it is continuous at $x=1$. At $x=1$, the left-hand limit is $3(1)-2=1$, and the right-hand limit is $1^2-3(1)+a=a-2$. For continuity, we need $1=a-2$, so $a=3$. The correct choice is ③.

$\boxed{3}$