If $f(x)$ is continuous in the interval $[a,b]$, then $f(x)$ is bounded on $[a,b]$.

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Lesson Parent: 
Properties of Continuous Functions
Problem: 

If $f(x)$ is continuous in the interval $[a,b]$, then $f(x)$ is bounded on $[a,b]$.

Answer: 

When $f(x)$ is continuous in the interval $[a,b]$, then $f(x)$ is bounded on $[a,b]$.

SAMPLE SOLUTION

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Solution: 

Recall the definition of bounded and unbounded functions.

First, we know that there is closed interval around $a$ where $f(x)$ is bounded. Let's call the closed interval <Sign in to see all the formulas>. In the proof we can choose <Sign in to see all the formulas> to be as small as we want. We make sure to choose an <Sign in to see all the formulas> so that <Sign in to see all the formulas>.

Now since $f(x)$ is also continuous at $x_1$, there is a closed interval around $x_1$ for which $f(x)$ is bounded. Since we know that everything to the left of $x_1$ is bounded we say we have an additional bounded interval <Sign in to see all the formulas>. (Again <Sign in to see all the formulas>.) So now we know that not only is $f(x)$ bounded on <Sign in to see all the formulas> but it is bounded on the larger interval <Sign in to see all the formulas>. We can keep doing this to get an infinite set of numbers <Sign in to see all the formulas> where $f(x)$ is bounded on the intervals <Sign in to see all the formulas>, <Sign in to see all the formulas>, <Sign in to see all the formulas>.

The set <Sign in to see all the formulas> has a least upper bound that we call $B$. We assert that $B=b$. If $B$ were less than $b$, say <Sign in to see all the formulas> then because $f(x)$ is continuous at <Sign in to see all the formulas> then there is some value of $x$ in <Sign in to see all the formulas> that is greater than <Sign in to see all the formulas>. This means that $f(x)$ is bounded on $[a,x]$ for all <Sign in to see all the formulas>. i.e. $f(x)$ is bounded on $[a,b)$.

So what happens at $x=b$? Using the same argument that we did at the beginning, we know that there is closed interval around $b$ where $f(x)$ is bounded. Let's call the closed interval <Sign in to see all the formulas> where <Sign in to see all the formulas>.

The union of the two intervals $[a,b)$ and <Sign in to see all the formulas> is $[a,b]$. Since $f(x)$ is bounded on the two intervals it is bounded on $[a,b]$.