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I'm a graduate student of mathematics in Germany.


Apr
10
asked How could I have found the closed form of $\sum_{k=1}^n \frac{k}{(k+1)!}$ in advance?
Apr
9
comment Definition of field extension
You’ll have to keep track of all the morphisms and all the different fields. Other than that, it’s a generalization and has all the classical advantages and disadvantages of generalizations: Some definitions, concepts and proofs become clearer, and with others you have to be more careful. For example, naturally a intermediate extension of $E/F$ would be an extension $L/F$ such that $E/L$ is an extension, too. Now, discerning these intermediate extensions becomes more subtle. And how can you talk about the “number” of intermediate extensions? One has to be more careful.
Apr
9
comment Continuity of the sum of continuous functions
@Jens Well, you can redo the proof that the composition of continuous functions is continuous for your special case. This is sort of only using the preimage definition.
Apr
9
comment Continuity of the sum of continuous functions
This is only true if $f$ is continuous. So show instead that the absolute value $ℝ → ℝ,\; x ↦ |x|$ is continuous. (Well, this actually is almost the definition of the topology of $ℝ$.) Then $|f| = |·| ∘ f$.
Apr
9
comment Determining injectivity and surjectivity
Try to write an integer in two different ways using the functions respectively – if this is possible, the respective function is not injective. Try to find an integer which you cannot write using the functions respectively – if you find one, the respective function is not surjective. Also, think of unique prime factorization.
Apr
9
revised Determining injectivity and surjectivity
added 10 characters in body
Apr
9
revised Determining injectivity and surjectivity
edited tags
Apr
9
comment Determining injectivity and surjectivity
You really need to know the domains and codomains to decide this.
Apr
9
comment Continuity of the sum of continuous functions
Prove $B(a,δ) × B(b,δ) ⊂ (+)^{-1}[B(\underline{a+b},ε)]$ instead.
Apr
8
comment Continuity of the sum of continuous functions
@Jens I didn’t characterize the preimage of an open ball around $c$, but I instead said that any point in this preimage has an open neighbourhood within the preimage. The cartesian products of open sets in $ℝ$ is open in $ℝ × ℝ$. I firmly believe you can show the continuity of the absolute value yourself using the same trick. Ask again if you couldn’t manage to do so.
Apr
8
revised Prove that $f$ is constant if $f'=0$
added 13 characters in body
Apr
8
answered Prove that $f$ is constant if $f'=0$
Apr
8
comment Prove that $f$ is constant if $f'=0$
You can do this by showing $\exp(z) \exp(c-z) = \exp(z + (c-z)) = \exp(c)$, using the functional equation $\exp(a+b) = \exp(a)\exp(b)$ (i.e. $\exp \colon ℂ → ℂ^×$ is a group homomorphism). The question statement is true nevertheless (assuming domains are meant to be connected), and I think there are several generalisations which should be easy to prove.
Apr
8
comment Prove that $f$ is constant if $f'=0$
Why do you want to show that and how far did you come?
Apr
8
comment Continuity of the sum of continuous functions
@Jens Is that okay?
Apr
8
revised Continuity of the sum of continuous functions
explained in more detail as requested in a comment
Apr
8
answered Continuity of the sum of continuous functions
Apr
8
comment Possible logical meanings of mathematical operations
In $\mathbb{F}_2 = ℤ/2ℤ$, if you interpret zero as false and one as true, “$+$” (and “$-$”) corresponds to exclusive or and “$·$” corresponds to and.
Apr
7
comment Does this topology on the dual have a name
Wht did you choose $ℂ$ to be the codomain of the $p_x$?
Apr
7
comment Are there non-surjective homeomorphisms?
By that phrase, they just mean homeomorphisms $X → X$, i.e. whose domain and codomain is $X$.