Chapter 3 Continuity
Definition 96.
We say that a function \(f\) is continuous at a point \(x\) in its domain (or at the point \((x,f(x))\)) if, for any open interval \(S\) containing \(f(x)\text{,}\) there is an open interval \(T\) containing \(x\) such that if \(t\in T\) is in the domain of \(f\text{,}\) then \(f(t)\in S\text{.}\)
Definition 97.
We say a function \(f\) is continuous if it is continuous at every point in its domain.
Let’s show that this definition of continuity behaves the way we expect from calculus. The first example will make it particularly easy to find \(T\text{,}\) given \(S\text{.}\)
Example 98.
Show that \(f(x) = x\) is continuous for all \(x\in \R\text{.}\)
Example 99.
Pick one: show that either \(f(x) = 2x\) or \(g(x) = x+3\) is continuous for all \(x\in \R\text{.}\)
Problem 100.
Prove that any linear function is continuous for all \(x\in \R\text{.}\)
Problem 101.
Let \(f(x) = 1\) if \(x\in [0,1]\text{,}\) and \(f(x) = 0\) otherwise. Determine and prove exactly for which \(x\) this function is continuous.
Problem 102.
Let \(f(x) = x^2\text{.}\) Show that \(f\) is continuous at \(x = 2\text{.}\) (You may assume that square roots exist.)
Problem 103.
Define a function \(g : \set{0}\to \R\) by \(g(0) = 0\text{.}\) Show that \(g\) is continuous at \(x=0\text{.}\)
Problem 104.
Let \(f(x) = 1\) if \(x\in \Q\text{,}\) and \(f(x) = 0\) otherwise. Find all points \(x\) where \(f\) is continuous.
It turns out we can connect continuity to sequences, too.
Definition 105.
We say that a function \(f\) is sequentially continuous at a point \(x\) if, for every sequence \(\set{x_i}_{i=1}^\infty\) (in the domain of \(f\)) converging to \(x\text{,}\) it is also true that \(\set{f(x_i)}_{i=1}^\infty\) converges to \(f(x)\text{.}\)
Problem 106.
If a function \(f\) is continuous at \(x\text{,}\) it is also sequentially continuous at \(x\text{.}\)
Problem 107.
If a function \(f\) is sequentially continuous at \(x\text{,}\) it is also continuous at \(x\text{.}\)
Hint.
Perhaps try the contrapositive?
So these concepts are equivalent on the real numbers. However, we still distinguish between them, since this isn’t the case for all sets. See, for instance:
The sequential way of thinking of continuity makes proving some basic facts easier. Be clear on the logic in the next two, and don’t forget what we’ve already proved or stated about sequence convergence.
Problem 108.
If \(c\in\R\) is a constant and \(f\) and \(g\) are continuous at a point \(x\text{,}\) then so are the functions \(cf\) and \(f+g\text{.}\)
Problem 109.
If \(f\) and \(g\) are continuous at a point \(x\text{,}\) so is the function \(fg\text{.}\)
Example 110.
The power function \(p(x) = x^n\) is continuous at every \(r\in\R\) for all \(n\ge 1\text{.}\)
Example 111.
All polynomials in \(x\) are continuous for all real numbers \(r\) (here, in the full domain \(\R\)).
Before finally moving on to calculus proper, we’ll deal briefly with two foundational theorems about \(\R\) and continuous functions that have everything to do with completeness.
The Extreme Value Theorem.
Let \(I\) be a closed interval and suppose that \(f\) is a function which is continuous on \(I\text{.}\) Then there exists \(x_M \in I\) such that \(f(x_M) \ge f(x)\) for all \(x\in I\text{.}\)
The Intermediate Value Theorem.
Suppose \(f\) is a continuous function on a closed interval \(I = [a,b]\text{,}\) and that \(f(a) \lt 0 \lt f(b)\text{.}\) Then there is a point \(x\in [a,b]\) such that \(f(x) = 0\text{.}\)
Problem 112.
Suppose \(f\) is a continuous function on a closed interval \(I = [a,b]\) and that \(f(a) \lt c \lt f(b)\text{.}\) Then there is a point \(x\in [a,b]\) such that \(f(x) = c\text{.}\)
Hint.
Feel free to use the IVT; we’ll prove it in
Problem 118.
Note that this is also true if \(f(b) \lt c \lt f(a)\text{.}\) One way to get a handle on these facts is to find counterexamples when the hypotheses are not satisfied.
Problem 113.
Find a continuous function on \(\R\) which does not satisfy the conclusions of the Extreme Value Theorem; do the same for a discontinuous function on \([0,1]\text{.}\)
Problem 114.
Find a continuous function on \([0,1]\cup [2,3]\) with \(f(0) = -1\text{,}\) \(f(3) = 1\) which does not satisfy the conclusions of the Intermediate Value Theorem; do the same for a discontinuous function on \([0,1]\) where \(f(0) = -1\text{,}\) \(f(1) = 1\text{.}\)
Proving the EVT is most often done via the Bolzano-Weierstrauss Theorem and the sequential definition of continuity (or other definitions); it’s rather tricky to do it using only the supremum definition. Here is one piece of the proof (this piece actually applies to non-continuous functions).
Problem 115.
Let \(f\) be a function on \([0,1]\text{,}\) and assume that the image \(f([0,1])\) has a supremum. Show there is a sequence of points \(\set{x_i}_{i=1}^\infty\) in \([0,1]\) such that \(\set{f(x_i)}_{i=1}^\infty\) converges to that supremum.
Example 116.
Explain why that’s not enough, by itself, to prove the whole Extreme Value Theorem.
Proving the IVT, on the other hand, can definitely be done using just the set definitions of completeness and continuity. The next problem should get your creative juices flowing as to how to find the \(x\) such that the IVT holds for that \(x\text{.}\) Be careful to show that your \(x\) exists!
Problem 117.
Let \(f\) be a function on \([0,1]\) such that \(f(0) = -1, f(1) = 1\text{,}\) and \(f([0,1]) = \set{-1,1}\text{.}\) Show (without using the IVT) that there is an \(x\in [0,1]\) such that \(f\) is not continuous at \(x\text{.}\)
Problem 118.
Prove the Intermediate Value Theorem.
The Intermediate and Extreme Value Theorems combine in a deep way. If we had time for just a bit more topology (maybe at the end of the semester?), we could have introduced the words compact and connected; these more advanced concepts are necessary to describe the proper generalization of the following result for more general spaces than \(\R\text{.}\)
Problem 119.
The image of a closed interval under a continuous function is either a closed interval or a single point.
Problem 120.
Find a function \(f\) on a closed interval \(I\) such that \(f(I)\) is also a closed interval, but for which \(f\) is not continuous for some \(x\in I\text{.}\)
Find a continuous function \(g\) on a closed interval \(I = [a,b]\) such that \(g(a) = g(b)\) but \(g(I)\) is not (just) the single point \(g(a)\text{.}\)
Finally, find a function \(h\) on a closed interval \(I\) such that \(h(I)\) is not a closed interval.
