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Define the sum of the non-negative numbers $\{r_s \mid s \in S\}$ $S$ uncountable to be

$$\sup _{D \subseteq S} \sum _{d \in D} r_d$$

($D$ being finite), which exists if this supremum is finite.

Define a point function to be a function from $[0, 1]$ to $\mathbb R$ that is $0$ everywhere except for a single point, where it takes a positive value.

Suppose we have an uncountable family of point functions $f_r: [0, 1] \to \mathbb R$ indexed by $r \in [0, 1]$. Define the pointwise sum function $S[a, b]: [0, 1] \to \mathbb R$ as

$$S[a, b] (x) = \sum _{r \in [a, b]} f_r (x) \ .$$

It can be shown that if $S[0, 1]$ is well defined, then so is $S[0, a]$ for any $a$ such that $0 \le a < 1$.

Prove or disprove that if $S[0, 1]$ is well defined, then for Lebesgue almost every $a \in [0, 1]$ the function $S[0, a]$ is discontinuous at at least one point.

Define the sum of the non-negative numbers $\{r_s \mid s \in S\}$ $S$ uncountable to be

$$\sup _{D \subseteq S} \sum _{d \in D} r_d$$

($D$ being finite), which exists if this supremum is finite.

Define a point function to be a function from $[0, 1]$ to $\mathbb R$ that is $0$ everywhere except for a single point, where it takes a positive value.

Suppose we have an uncountable family of point functions $f_r: [0, 1] \to \mathbb R$ indexed by $r \in [0, 1]$. Define the pointwise sum function $S[a, b]: [0, 1] \to \mathbb R$ as

$$S[a, b] (x) = \sum _{r \in [a, b]} f_r (x) \ .$$

It can be shown that if $S[0, 1]$ is well defined, then so is $S[0, a]$ for any $a$ such that $0 \le a < 1$.

Assume that if $S[0, 1]$ is well defined. Does it follow that for Lebesgue almost every $a \in [0, 1]$ the function $S[0, a]$ is discontinuous at at least one point?

Define a sum of non-negative numbers indexed by the uncountable set S to be

Sup (D finite subset of S) sum (d in D) r_d,

  which exists if this supremum is finite.

Define a point function to be a function from [0, 1] to R that is zero everywhere except for a single point, where it takes a positive value.

Suppose we have an uncountable family of point functions f_r: [0, 1] -> R indexed by r in [0, 1]. Define the pointwise sum function S[a, b]: [0, 1] -> R as S[a, b] (x) = Sum (r in [a, b]) f_r (x).

It can be shown that if S[0, 1] is well defined, then so is S[0, a] for any a such that 0 <= a < 1.

Prove or disprove that if S[0, 1] is well defined, then for Lebesgue almost every a in [0, 1], the function S[0, a] is discontinuous at at least one point.

Define the sum of the non-negative numbers $\{r_s \mid s \in S\}$ $S$ uncountable to be

$$\sup _{D \subseteq S} \sum _{d \in D} r_d$$

($D$ being finite), which exists if this supremum is finite.

Define a point function to be a function from $[0, 1]$ to $\mathbb R$ that is $0$ everywhere except for a single point, where it takes a positive value.

Suppose we have an uncountable family of point functions $f_r: [0, 1] \to \mathbb R$ indexed by $r \in [0, 1]$. Define the pointwise sum function $S[a, b]: [0, 1] \to \mathbb R$ as

$$S[a, b] (x) = \sum _{r \in [a, b]} f_r (x) \ .$$

It can be shown that if $S[0, 1]$ is well defined, then so is $S[0, a]$ for any $a$ such that $0 \le a < 1$.

Prove or disprove that if $S[0, 1]$ is well defined, then for Lebesgue almost every $a \in [0, 1]$ the function $S[0, a]$ is discontinuous at at least one point.

Define a sum of non-negative numbers indexed by the uncountable set S to be

Sup (D finite subset of S) sum (d in D) r_d,

which exists if this supremum is finite.

Define a point function to be a function from [0, 1] to R that is zero everywhere except for a single point, where it takes a positive value.

Suppose we have an uncountable family of point functions f_r: [0, 1] -> R indexed by r in [0, 1]. Define the pointwise sum function S[a, b]: [0, 1] -> R as S[a, b] (x) = Sum (r in [a, b]) f_r (x).

It can be shown that if S[0, 1] is well defined, then so is S[0, a] for any a such that 0 <= a < 1.

Conjecture: For Lebesgue almost every a in [0, 1], the function S[0, a] is discontinuous at at least one point.

Define a sum of non-negative numbers indexed by the uncountable set S to be

Sup (D finite subset of S) sum (d in D) r_d,

which exists if this supremum is finite.

Define a point function to be a function from [0, 1] to R that is zero everywhere except for a single point, where it takes a positive value.

Suppose we have an uncountable family of point functions f_r: [0, 1] -> R indexed by r in [0, 1]. Define the pointwise sum function S[a, b]: [0, 1] -> R as S[a, b] (x) = Sum (r in [a, b]) f_r (x).

It can be shown that if S[0, 1] is well defined, then so is S[0, a] for any a such that 0 <= a < 1.

Prove or disprove that if S[0, 1] is well defined, then for Lebesgue almost every a in [0, 1], the function S[0, a] is discontinuous at at least one point.