Continuity

Topology/All notes11 Aug 2026

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Theorem 0.1 — continuity

Let $𝐴\subset ℝ$, a function $𝑓:𝐴\to ℝ$ is continuous if and only if $\forall \text{open set}𝑈\subset ℝ,\exists \text{some open set}𝑉\subset ℝ,\text{such that}{𝑓}^{-1}\left(𝑈\right)=𝑉\cap 𝐴$.

Proof

Let ${𝑥}_0\in 𝐴$, $𝜀>0$. Setting $𝑈\subset ℝ=\left(𝑓\left({𝑥}_0\right)-𝜀,𝑓\left({𝑥}_0\right)+𝜀\right)$ and ${𝑓}^{-1}\left(𝑈\right)=𝑉\cap 𝐴$, where $𝑉\subset ℝ$. Then ${𝑥}_0\in 𝑉$. Since $𝑉$ is an open set, there exists $\left({𝑥}_0-𝛿,{𝑥}_0+𝛿\right)\subset 𝑉$. Because ${𝑥}_0\in 𝑉\cap 𝐴$, $𝑉\cap 𝐴$ is non-empty. Hence, $\forall 𝑥\in ℝ\text{which makes}|𝑥-{𝑥}_0|<𝛿,|𝑓\left(𝑥\right)-𝑓\left({𝑥}_0\right)|<𝜀$. Thus, $\forall 𝜀>0,\exists 𝛿>0,|𝑥-{𝑥}_0|<𝛿\Rightarrow |𝑓\left(𝑥\right)-𝑓\left({𝑥}_0\right)|<𝜀$.
The converse is similar and is left to the reader.⁠∎

View Typst source

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#import "@preview/noteworthy:0.4.0": *
#import "/themes/site/notes.typ": *

#show: tylenotes.with(
  title: "Continuity",
  date: "2026-08-11",
  tags: ("topology", "lean"),
  summary: "An equivalent characterization of continuity for functions on A ⊆ ℝ: the preimage of every open set is open in the subspace topology.",
)

#theorem(title: "continuity")[
  Let $A subset RR$,
  a function $f:A -> RR$ is continuous if and only if
  $ forall "open set" U subset RR, exists "some open set" V subset RR, "such that" f^(-1)(U)=V inter A$.
]

#proof[
  Let $x_0 in A$, $epsilon > 0$.
  Setting $U subset RR = (f(x_0)-epsilon, f(x_0)+epsilon)$ and $f^(-1)(U) = V inter A$,
  where $V subset RR$.
  Then $x_0 in V$.
  Since $V$ is an open set, there exists $(x_0-delta, x_0+delta) subset V$.
  Because $x_0 in V inter A$, $V inter A$ is non-empty. 
  Hence, $forall x in RR "which makes" |x - x_0| < delta, |f(x)-f(x_0)|<epsilon$.
  Thus, $forall epsilon > 0, exists delta > 0, |x - x_0| < delta => |f(x)-f(x_0)| < epsilon$. \
  The converse is similar and is left to the reader.
]