Nuprl Lemma : function-is-continuous
∀I:Interval. ∀f:I ⟶ℝ. ((∀x,y:{t:ℝ| t ∈ I} . ((x = y) ⇒ (f[x] = f[y]))) ⇒ f[x] continuous for x ∈ I)
Proof
Definitions occuring in Statement :
continuous: f[x] continuous for x ∈ I,
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
req: x = y,
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
continuous: f[x] continuous for x ∈ I,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
rfun: I ⟶ℝ,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
and: P ∧ Q,
sq_stable: SqStable(P),
squash: ↓T,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
ifun: ifun(f;I),
real-fun: real-fun(f;a;b),
subinterval: I ⊆ J ,
top: Top
Latex:
\mforall{}I:Interval. \mforall{}f:I {}\mrightarrow{}\mBbbR{}.
((\mforall{}x,y:\{t:\mBbbR{}| t \mmember{} I\} . ((x = y) {}\mRightarrow{} (f[x] = f[y]))) {}\mRightarrow{} f[x] continuous for x \mmember{} I)
Date html generated:
2020_05_20-PM-00_04_46
Last ObjectModification:
2020_01_08-AM-10_38_36
Theory : reals
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