Question

Answer the following True or False:

If lim f(x) = – 3 then f(x) is continuous at x = – 2.
2 - 2

Answer 1

False; consider as a counterexample the function f : ℝ→ℝ defined by

`f(x)=\begin{cases}-3&\text{for }x\\eq2\\0&\text{for }x=2\end{cases}`

Clearly f approaches -3 as x gets closer to -2, but neither limit from either side is equal to the function's value at x = 2 (that is, -3 ≠ 0), so f is not continuous.