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Prove that if $f(x)$ is uniformly continuous on a set $U$ and $a$ is inside $U$ , then $f(x)$ is continuous at $a$ .

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Lesson Parent:

Uniform Continuity

Problem:

Prove that if $f(x)$ is uniformly continuous on a set $U$ and $a$ is inside $U$ , then $f(x)$ is continuous at $a$ .

Answer:

It is true that if $f(x)$ is uniformly continuous on a set $U$ and $a$ is inside $U$ , then $f(x)$ is continuous at $a$ .

Solution:

Recall our definition of uniform continuity which says that for every <Sign in to see all the formulas> there exists a <Sign in to see all the formulas> such that for every $x$ and $z$ in $U$ , if <Sign in to see all the formulas>, then <Sign in to see all the formulas>. Let's make $z=a$ so that we now have for every $x$ and $a$ in $U$ , if <Sign in to see all the formulas>, then <Sign in to see all the formulas>.

We are basically done, but there are two subtle points that should probably be made just to make sure we are complete in our discussion. First, the definition of continuity says that for every <Sign in to see all the formulas> there exists a <Sign in to see all the formulas> such that if <Sign in to see all the formulas>, then <Sign in to see all the formulas>. The difference between what we have and the definition of continuity is that we know if <Sign in to see all the formulas>, then <Sign in to see all the formulas>, but the definition of continuity is that if <Sign in to see all the formulas>, then <Sign in to see all the formulas>. The difference is that what we currently have is an $x$ such that <Sign in to see all the formulas> and the definition of continuity has $x$ such that <Sign in to see all the formulas>. Recall that <Sign in to see all the formulas> means the union <Sign in to see all the formulas> while <Sign in to see all the formulas> is the set <Sign in to see all the formulas>. The point here is that <Sign in to see all the formulas> is a subset of <Sign in to see all the formulas>. In other words, since <Sign in to see all the formulas> is true for <Sign in to see all the formulas> it is also true for <Sign in to see all the formulas>. Therefore, we know that for every <Sign in to see all the formulas> there exists a <Sign in to see all the formulas> such that if <Sign in to see all the formulas>, then <Sign in to see all the formulas>. Since we know this is true and is the exact definition of continuity then we know that $f(x)$ is continuous at $a$ .

Second, the reason that we say $a$ must be inside $U$ is because that is how we define function limits. See this lesson for more detail. The point is that we demand $a$ to be inside the domain to match with the definition of function limits.