Defining Continuity at a Point AB & BC

A function is continuous at $x = a$ when three things hold: $f(a)$ is defined, $\lim_{x\to a} f(x)$ exists, and the two are equal. All three are independent, and each can fail while the others hold. Compactly the definition is the single equation $\lim_{x\to a} f(x) = f(a)$, which quietly asserts that both sides exist. Polynomials are continuous everywhere, and rational, radical, exponential, logarithmic, and trigonometric functions are continuous on their domains.

The characteristic error is checking part of the test and stopping, most often verifying that the limit exists and concluding continuity without comparing it to the function value. That is exactly the case a hole with a plotted dot elsewhere is built from. The related error is confusing the limit with the value in the first place, so the comparison never happens. When solving for a constant that forces continuity, matching the two one-sided limits is necessary but not sufficient: the function value at the join has to equal them too.