The following statements about the deleted limit are equivalent. (a) b = lim_c f exists. (b) For each ε > 0 there is δ > 0 such that ‖f(x) − b‖ < ε whenever x ∈ D and 0 < ‖x − c‖ < δ. (c) For every sequence (xₙ) in D with xₙ ≠ c and c = lim(xₙ), we have b = lim(f(xₙ)).
(b) 是可以動筆的版本，那個 0 < ‖x − c‖ 就是「不看 c 自己」的具體寫法。(c) 把檢查交給數列，而它最好用的方向是否定——找出兩條都跑向 c 而像跑向不同地方的數列，極限就被判掉了。
Proof. (a) ⟹ (b): given ε > 0, apply the definition with V the ball of radius ε about b. The resulting neighborhood U of c contains a ball of radius δ > 0 about c, and this δ serves. (b) ⟹ (c): let (xₙ) be as in (c) and let ε > 0. Take δ from (b) and then K with ‖xₙ − c‖ < δ for n ≥ K. Since xₙ ≠ c, we get 0 < ‖xₙ − c‖ < δ and hence ‖f(xₙ) − b‖ < ε.
(c) ⟹ (b): suppose (b) fails. Then there is ε₀ > 0 such that for each n the choice δ = 1/n fails, giving xₙ ∈ D with 0 < ‖xₙ − c‖ < 1/n and ‖f(xₙ) − b‖ ≥ ε₀. Then xₙ ≠ c and xₙ → c, while (f(xₙ)) does not converge to b; so (c) fails. Finally (b) ⟹ (a): a ball is a neighborhood, and conversely every neighborhood of b contains a ball about b by 9.7, so the two formulations match on both sides.
這張圖在說 25.3 三種說法量的是同一個範圍：ε-δ 版把它寫成兩條虛線之間、而且挖掉中心那一點（空心圈）的兩段粗線；數列版則把同一件事說成「任何一條跑向 c 而不踩上 c 的點列，遲早整條落在那兩段裡」。挖掉中心正是 deleted 版與 non-deleted 版的唯一分野。
25.4 THEOREM
The corresponding statements about the non-deleted limit are equivalent: b = Lim_c f exists; for each ε > 0 there is δ > 0 with ‖f(x) − b‖ < ε whenever x ∈ D and ‖x − c‖ < δ; and b = lim(f(xₙ)) for every sequence (xₙ) in D with c = lim(xₙ).
Proof. Delete every occurrence of the clauses x ≠ c and xₙ ≠ c from the proof of 25.3. Each step remains valid: the only place they were used was to license the strict inequality 0 < ‖x − c‖, which is no longer required.
Let c be a cluster point of D that belongs to D. Then the following are equivalent. (a) f is continuous at c. (b) lim_c f exists and equals f(c). (c) Lim_c f exists.
連續 ＝ 極限存在而且落在該取的位置。(c) 特別值得玩味：non-deleted 版只要「存在」就夠了，不必額外要求它等於 f(c)——因為 c 本身被算進來，它的值自動被鎖在極限上。這也解釋了為什麼許多文獻直接用 non-deleted 極限來定義連續。
走 (a) ⟹ (c) ⟹ (b) ⟹ (a) 一圈。前提 c ∈ D 在整段裡反覆用到——沒有它，f(c) 根本不存在。
證明計畫 · 由所求想起 (a)⟹(c)：連續的敘述逐字就是 non-deleted 極限等於 f(c) 的敘述，直接讀出來即可。 (c)⟹(b)：所求是把那個「存在」的極限值釘在 f(c) 上。關鍵是 c 自己也在被檢查的範圍內——於是 f(c) 落在每一個 V 裡，而任意小的 V 只框得住一個點。再用 25.2(b) 換成 deleted 版。 (b)⟹(a)：deleted 版只漏了 x = c 那一格，而那一格自動成立。
Proof. (a) ⟹ (c): continuity at c says that for every neighborhood V of f(c) there is a neighborhood U of c with f(x) ∈ V for all x ∈ U ∩ D. That is precisely the statement Lim_c f = f(c).
(c) ⟹ (b): let b = Lim_c f. For every neighborhood V of b the corresponding U contains c, and c ∈ D, so f(c) ∈ V. As V may be taken arbitrarily small, f(c) = b. By 25.2(b) the deleted limit also exists and equals b = f(c).
這一步是全證明的重點。f(c) 落在每一個 V 裡，而 V 可以取成任意小的球——若 f(c) ≠ b，取半徑小於兩者距離的球就把 f(c) 排除在外，矛盾。這正是 non-deleted 版「只要存在就夠」的機制：c 在檢查範圍內，它的值就被綁在極限上，沒有第二種可能。
(b) ⟹ (a): let V be a neighborhood of f(c) and take U from the deleted limit, so that f(x) ∈ V for x ∈ U ∩ D with x ≠ c. For x = c we have f(c) ∈ V because V is a neighborhood of f(c). Hence f(x) ∈ V for every x ∈ U ∩ D, which is continuity at c.
這一格是邊界簿記，可是它說明了 (b) 為什麼要多寫「而且等於 f(c)」。補上 x = c 那一格時，用的正是「極限值就是 f(c)」這個假設——若 deleted limit 存在卻等於別的數，這一格就補不起來，§25-1 例 1 第 2 步就是現成的失敗案例。