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Calculus

Continuity

A function is continuous at a number \(a\) if 

\[ \lim_{x\ \to a }(f(x)) = f(a)\]

The above statement looks like just one thing must happe for a function to be continuous at \(a\), but to check if a function is continuous, we need three conditions satisfied:

  1. \(f(a)\) must be defined (\(a\) is in the domain of \(f\)).
  2. \(\lim_{x \to a}f(x)\) exists (the right and left limits must agree).
  3. \(\lim_{x \to a}f(x) = f(a)\).

Take a look at the picture below and try to determine if the function $f$ is continuous at \(x = 1,2,3\) and \(4\). If the functions is not continuous at one of the numbers, make a list of all of the above above conditions that the function does not satisfy. 

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The solution is discussed in the video below.

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