The discussion focuses on proving the continuity of two functions: f(x) = (x + 2x^3)^4 at a = -1 and f(x) = (2x + 3)/(x - 2) at x = 2. For the first function, it is established that since it is a polynomial, it is continuous for all real numbers, including at -1, where the limit equals the function value. In the second case, the function is shown to be continuous everywhere except at x = 2 due to a division by zero, but the limit as x approaches 2 is evaluated to be 2, which matches the function's value at that point. The properties of limits and the definition of continuity are key to these proofs. Overall, the continuity of both functions is effectively demonstrated through limit evaluation.