Question

a) Every infinite set is an open set.

b)A necessary condition for a function f to be integrable is that it is continuous.


true or false

Answer 1

Answer on Question #84078 – Math – Real Analysis

Question

a) Every infinite set is an open set.

b) A necessary condition for a function $f$ to be integrable is that it is continuous.

true or false?

Solution

a) False. A counterexample can be the set of integer numbers $\{...,-2,-1,0,1,2,...\}$ on the real axis, which is infinite as a set of points but not open (it is closed, in fact), or intervals of type $[a,b)$, $(a,b]$, $[a,b]$ with $a < b$, which are infinite as sets of points, but not open.

b) False. The function $f(x)$ such that $f(x) = 0$ for $x \leq 0$, and $f(x) = 1$ for $x > 0$ is integrable on every finite interval, but it is not continuous at $x = 0$.

Answer: a) false; b) false.

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