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Prove that continuous functions are integrable.

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Problem:

Prove that continuous functions are integrable. Prove that if $f(x)$ is continuous on the closed interval $[a,b]$ , then $f(x)$ is integrable there.

Answer:

It is true that continuous functions are integrable.

Solution:

Now let's assume that we have a partition $P$ such that <Sign in to see all the formulas> (remember that $||P||$ is the partition's norm). So let's focus our attention on an arbitrary interval of the partition and denote it by <Sign in to see all the formulas>. Now because $f(x)$ is continuous on the interval <Sign in to see all the formulas> we know from the Extreme-Value Theorem that there are two points $e_i$ and $E_i$ in <Sign in to see all the formulas> where $f(x)$ will take in its minimum and maximum values which we denote by $m_i$ and $M_i$ , respectively. In other words, <Sign in to see all the formulas> and <Sign in to see all the formulas>.

So now let's form our upper and lower sums and subtract to get

But because we know that <Sign in to see all the formulas> we have

Recall from this lesson that one of our three equivalent definitions of the definite integral is the following: for every <Sign in to see all the formulas> there exists a <Sign in to see all the formulas> such that for every partition $P$ , if <Sign in to see all the formulas>, then <Sign in to see all the formulas>. But we have just proven this! Therefore, $f(x)$ is integrable. The proof is now complete.