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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.

The solution is discussed in the video below.