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Results tagged with continuity
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user 1684451
Intuitively, a continuous function is one where small changes of input result in correspondingly small changes of output. Use this tag for questions involving this concept. As there are many mathematical formalizations of continuity, please also use an appropriate subject tag such as (real-analysis) or (general-topology)
4
votes
1
answer
155
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Properties of continuous functions on $(0,\infty)$ with $f(x) = f(2x)$ for all $x$.
Using the sequential definition of continuity, I got $f(x) = f(0)$ for each $x\in(0,\infty)$ but this is bogus since the domain of $f$ does not include $0$. From here, I am confused. …
- 342
2
votes
2
answers
104
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Continuity of $\sum_{n = 1}^{\infty} e^{-nx}\sin{nx}$.
I was doing a problem where I was asked to show the continuity of
$$\sum_{n = 1}^{\infty} e^{-nx}\sin{nx},$$
for $x>0$. …
- 342
1
vote
1
answer
57
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A map defined on a compact domain is continuous if and only if its graph is compact. [duplicate]
Problem: If $E$ is a compact subset of a metric space $X$, and $f$ is a map defined on $E$ to a metric space $Y$, then prove that the graph of $f$, denoted by $G(f)$ and defined as the set
$$\{(x,f(x) …
- 342