KindiakMath

Formalising Continuity

While the usual study of calculus involves derivatives and integrals, continuity plays an essential role in make sure these notions behave as expected.

We give the usual definition of continuity here, prove the continuity of polynomials, and state one visually intuitive theorem that takes surprisingly more effort than expected to prove. We will need this to prove the fundamental theorem of calculus.

Definition 1. Let c be a real number f be a real valued function defined on (c-r,c+r) for some r > 0. We say that f is continuous at x = c if

\displaystyle \lim_{x \to c} f(x) = f(c).

We say that f is continuous on a subset I \subseteq \mathbb R precisely when f is continuous at every c \in I.

Using limit laws, it can be shown that continuous functions remain continuous under addition, subtraction, and multiplication. Division also works so long as the divisor has a nonzero function value. Rather trivially, it is not hard to prove that (i) constant functions are continuous, and (ii) x is continuous in x. We prove (ii) for completeness.

Lemma. For any real c, the function f(x) = x is continuous at c.

Proof. Continuity can be reformulated in limits as

\displaystyle \lim_{x \to c} f(x) = f(c)\quad \iff \quad f(c+h) = f(c) + o(1),\quad h \to 0.

Setting f(x) = x,

f(c+h) = c+h = f(c) + o(1),

as required.

Let’s first define polynomials, whose continuity we aim to prove.

Definition 2. A non-zero real-valued function f is a polynomial in x with degree n \geq 0 if there exist real constants a_0,a_1,\dots,a_n such that for any real x,

\displaystyle f(x) = \sum_{k=0}^n a_k x^k = a_0 + a_1x + \cdots + a_nx^n.

We are now ready to prove that polynomials are continuous.

Theorem 1. Any polynomial is continuous on \mathbb R.

Proof. Since the product of two continuous functions is continuous, the product x^k of k copies of x is continuous as well. Since constant functions are continuous, all functions of the form a_kx^k are continuous. Finally, since the sum of continuous functions is continuous,

\displaystyle \sum_{k=0}^n a_k x^k = a_0 + a_1x + \cdots + a_nx^n,

is continuous.

We will state the extreme value theorem, which seems obvious, but whose proof we will omit.

Theorem 2 (Extreme Value Theorem). Let f : [a,b] \to \mathbb R be continuous. Then there exist c_1, c_2 \in [a,b] such that for any x \in [a,b],

f(c_1) \leq f(x) \leq f(c_2).

The extreme value theorem will be a crucial tool in studying differentiable functions, since the latter is continuous, and therefore inherit all properties of continuous functions.

Let’s establish a useful continuity result that will prove useful when discussing differentiation.

Theorem 3. Let f,g be real-valued functions. Suppose f is continuous at c and g is continuous at f(c). Then the composite function g \circ f defined by

(g \circ f)(x) := g(f(x))

is continuous at c.

Proof. Consider the computation

\begin{aligned} (g \circ f)(c + h) &= g(f(c + h))\\ &= g(f(c) + o(1)) \\ &= (g \circ f)(c) + o(1), \end{aligned}

as h \to 0.

—Joel Kindiak, 18 Oct 24, 1815H

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