The graph of a continuous $f : \mathbb{R} \to \mathbb{R}$ is closed in $\mathbb{R}^2$

Let $f : \mathbb{R} \to \mathbb{R}$ be continuous in the $\varepsilon$-$\delta$ sense of Continuity of $f : A \to \mathbb{R}$ at a point of $A$ and on $A$: the $\varepsilon$-$\delta$ condition, its agreement with $\lim_{x \to c} f(x) = f(c)$ at a limit point, and continuity at an isolated point, give $\mathbb{R}$ its usual topology (The absolute value makes $\mathbb{R}$ a metric space: $d(x,y) = |x-y|$ is a metric, its open balls are the intervals $(x-r, x+r)$, and it is unbounded, Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not) and give $\mathbb{R}^2 = \mathbb{R} \times \mathbb{R}$ the product topology (The product set $\prod_{i \in I} X_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space, For $n \ge 1$ the product topology on $n$ copies of the usual topology of $\mathbb{R}$ is the metric topology of $d_\infty$ on $\mathbb{R}^n$, and hence also of $d_1$ and $d_2$, so $\mathbb{R}^n$ as a product and $\mathbb{R}^n$ as a metric space are one space). Then the graph

$G_f \;=\; \{\, (x, f(x)) : x \in \mathbb{R} \,\}$

is closed in $\mathbb{R}^2$.

Every polynomial function is such an $f$, and so is every function built from continuous ones by the operations that preserve continuity; no further hypothesis on $f$ is needed, and in particular $f$ need not be bounded, monotone, or differentiable.

• What the $\varepsilon$-$\delta$ hypothesis becomes. The dictionary of Dictionary: for $A \subseteq \mathbb{R}$ with the metric $d(x,y) = |x-y|$, continuity and uniform continuity of $f : A \to \mathbb{R}$ agree with the metric-space notions, the Lipschitz and Hölder conditions are the metric ones instantiated, and a subset of $\mathbb{R}$ is compact in the open-cover sense of $\mathbb{R}$ exactly when it is a compact metric subspace and Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not is what lets a hypothesis stated with $\varepsilon$ and $\delta$ be fed to a theorem stated about topological spaces; there is one notion of continuity for a real function here, not two, and step 1.2 is where that is used.

• The converse fails. A discontinuous $\mathbb{R} \to \mathbb{R}$ may still have closed graph — the function equal to $1/x$ off $0$ does (FALSE: every function between topological spaces whose graph is closed in the product is continuous) — so "closed graph" is strictly weaker than "continuous" for real functions. What restores the equivalence is a compact Hausdorff codomain (A map into a compact space whose graph is closed is continuous; so for a compact Hausdorff codomain, continuity and closedness of the graph are equivalent).

• Nothing about $\mathbb{R}$ as a domain is used. The domain enters the argument only as an arbitrary topological space; the same proof gives a closed graph for a continuous map from any space into $\mathbb{R}$.