Question

Prove that the  \( f(x) = x^2 \) is continuous at \( x \neq 0 \).

Hint:

To prove continuity, check if the left-hand limit, right-hand limit, and function value are equal.

Answer 1

A 6266899d2bbfcb1799af2d57 is continuous at \( x = c \) if: \[ \lim_{x \to c} f(x) = f(c). \] For \( f(x) = x^2 \), we compute the left-hand and right-hand limits: \[ \lim_{x \to c} x^2 = c^2. \] Since \( f(c) = c^2 \), we get: \[ \lim_{x \to c} f(x) = f(c). \] Thus, \( f(x) = x^2 \) is continuous for all \( x \).