Equivalent formulations of function continuity
Let \( X \) be a subset of \( \mathbb{R} \), let \( f : X \to \mathbb{R} \) be a function, let \( x_0 \) be an element of \( X \). Then the following four statements are logically equivalent:
- \( f \) is continuous at \( x_0 \).
- For every sequence \( (a_n)_{n=0}^{\infty} \) consisting of elements of \( X \) which converges to \( x_0 \), the sequence \( (f(a_n))_{n=0}^{\infty} \) converges to \( f(x_0) \).
- For any real \( \varepsilon > 0 \) there exists a real \( \delta > 0 \) such that \( |f(x) - f(x_0)| < \varepsilon \) for all \(x \in X \) and \( |x - x_0| < \delta \)
Pseudo-proof
a) and b) correspond to a) and b) of Prop 9.3.9 (convergence of \( f \) to \( L \) is equivalent to all sequences converging to \( L \)); we just add the idea of continuity to replace \( L \) with \( f(x_0) \).
c) comes about by unwrapping the definition of function convergence giving us local ε-closeness; we just replace \( L \) with \( f(x_0) \).
Importance
Prop 9.3.9 was important in allowing us to determine fuction limits by switching to limits of sequences; this proposition (9.4.7), if the function in question is continuous, allows us to determine its limit, say at \( x_0 \) simply by evaluating \( f(x_0) \). This is far easier than working with the definition of function convergence directly.
In addition, it sets the stage for arithmetic preservation of continuity, which allows us to assert continuity for many functions, which in turn allows us to calculate the limits of functions easily.
[TODO: add a progress->line]