Function Limits VI: Continuous Functions
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A continuous function is one that can be drawn on a piece of paper without having to lift the pencil from the paper to draw it. For example,
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is an example of a continuous function.
However,
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is a function that cannot be drawn on a piece of paper without lifting your pencil. At $x=1$ the function has a value of 1 and immediately to the right of $x=1$ the function has a value of 5! This is definitely not a continuous function because we must lift our pencil off of the sheet at $x=1$ to continue drawing the function. Note that this function is continuous at $x=2$.
Furthermore, the functions
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are both discontinuous at $x=1$ because they are not defined there at all. Note that the second function is identical to our first example except at $x=1$.
The official mathematical definition of continuity is that a function is continuous at a point $x=a$ when
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We have always said that a function limit
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does not have to equal the function value at $x=a$. In other words, $L$ does not have to equal $f(a)$. But for a continuous function, <Sign in to see all the formulas>. This is how we can draw a picture of a function and not have to lift up our pencil.
Similar to one-sided limits, we can define one-sided continuity:
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and
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But we know that <Sign in to see all the formulas> if and only if <Sign in to see all the formulas> and <Sign in to see all the formulas>. Therefore, for continuous functions,
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The left-sided and right-sided function limits must equal $f(a)$ for a continuous function.
Finally, here are some important properties of continuous functions.
