Proof of the continuity of the sum of a uniformly convergent series of functions

Builds on Series of functions Product of continuous functions is continuous Uniform convergence on a set Uniform limit theorem

Proof

We apply the uniform limit theorem to the partial sums.

Let $S \subseteq \R$ , and let each $f_n \colon S \to \R$ be continuous. Let $s_N = \sum_{n=0}^{N} f_n$ be the partial sums of the series, and let $s$ be its sum.
We show by induction on $N$ that $s_N$ is continuous. For $N = 0$ , $s_0 = f_0$ is continuous.
If $s_N$ is continuous, then $s_{N+1} = s_N + f_{N+1}$ is continuous, since sums of continuous functions are continuous.
By the definition of a uniformly convergent series of functions, $s_N \to s$ uniformly on $S$ .
By steps 2, 3 and 4, the uniform limit theorem applies to the sequence $(s_N)$ . So $s$ is continuous on $S$ .

Idea

Each partial sum is a finite sum of continuous functions, so it is continuous. Uniform convergence of the series means uniform convergence of these partial sums. The uniform limit of continuous functions is continuous.

Reading path

To understand this page from the ground up, read these 76 pages in order. The foundations come first. The path takes the proofs Proof of the uniform limit theorem by an epsilon over three argument, Proof of the limit of a sum of sequences, Proof of the approximation property of the supremum, Proof of the epsilon-delta form of continuity, Proof that there is a natural number n with 1/n below any positive epsilon, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the limit of a product of sequences, Proof that the absolute value is multiplicative, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof that the reals are Archimedean from the completeness axiom, Proof that convergent sequences are bounded, Proof that continuity is equivalent to sequential continuity, Proof that sums and products of continuous functions are continuous.

Construction of the reals from Cauchy sequences concept
Element definition
Integers object
Limit point of a set definition
Ordered field definition
Proposition concept
Rational numbers object
Real numbers object
Sequence definition
Set bounded above definition
Set definition
Ring of integers statement
Limit of a function definition
Archimedean ordered field definition
Implication definition
Predicate definition
The reals form a field statement
Real sequence definition
Supremum definition
Ordered pair definition
Continuity at a point definition
The rationals form a field statement
Universal quantifier definition
Bounded sequence definition
Limit of a sequence definition
Series definition
Completeness axiom statement
Cartesian product definition
Continuous function definition
Subset definition
The rationals form an ordered field statement
Convergent sequence definition
Pointwise convergence definition
Sequential continuity definition
Binary relation definition
Order on the reals definition
Uniform convergence definition
Antisymmetric relation definition
Reflexive relation definition
Transitive relation definition
Interval definition
Uniform convergence on a set definition
Series of functions definition
Preorder definition
Proof of the epsilon-delta form of continuity proof
Partial order definition
Epsilon-delta definition of continuity statement
Total order definition
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
The reals form an ordered field statement
Absolute value definition
Proof of the approximation property of the supremum proof
Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Proof of the epsilon-N definition of the limit of a sequence proof
Approximation property of the supremum statement
Triangle inequality statement
Absolute value is multiplicative statement
Epsilon-N definition of the limit of a sequence statement
Proof that the reals are Archimedean from the completeness axiom proof
Proof of the limit of a sum of sequences proof
Proof of the uniform limit theorem by an epsilon over three argument proof
Proof that convergent sequences are bounded proof
The reals are Archimedean statement
Uniform limit theorem statement
Limit of a sum of sequences statement
Proof that there is a natural number n with 1/n below any positive epsilon proof
Convergent sequences are bounded statement
Proof of the limit of a product of sequences proof
There is a natural number n with 1/n below any positive epsilon statement
Limit of a product of sequences statement
Proof that continuity is equivalent to sequential continuity proof
Continuity is equivalent to sequential continuity statement
Proof that sums and products of continuous functions are continuous proof
Product of continuous functions is continuous statement
Proof of the continuity of the sum of a uniformly convergent series of functions proof

Where this leads

5 pages build on this one.

Tag	`K3XJ683740`
URI	`m2://K3XJ683740`
Kind	proof
Writing	W0 stub
MSC	40A30, 26A15
Proves	Continuity of the sum of a uniformly convergent series of functions
Uses	Series of functions, Product of continuous functions is continuous, Uniform convergence on a set, Uniform limit theorem
Changed	4 November 2026

Proof of the continuity of the sum of a uniformly convergent series of functions

Proof

Idea

Reading path

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