# Tagged Questions

Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, and integration through the fundamental theorem of calculus.

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### $f\in L^2([0,1])$ and $\int_0^1 t^n f(t)\,dt = (n+2)^{-1}$ for all $n=0,1,2,\ldots$. Is $f(t)=t$ a.e.?

I know how to prove the statement if $f$ is continuous, but the $L^2$ part is throwing me off. As far as I know, we can't use a version of Stone-Weierstrass, because $f$ isn't continuous. I'm pretty ...
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### Integrate function by partial derivative

I'm searching a $\phi(x,t)$ solution of a pde cauchy system, with $x\in[-1,1],t\in[0,T]$ I am able to know: a) $\phi(x,0)=-cos\left(\pi\left(x-0.85\right)\right)$ b) $\phi_x(x,t)$, $\forall t,x$ (...
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### Exponential equations in one variable for the reals [closed]

My father approached me yesterday and asked me if I could solve $$4^{x}+5^{x}=6^{x}$$ I countered by asking him over what set. He told me $R_{>0}$ So by using the intermediate value theorem it's ...
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### Symbol of differential operator and change of coordinates

Some time ago I posted the question about the change of coordinates in differential operator. Here is the relevant discussion Symbol of differential operator transforms like a cotangent vector The ...
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### Prove $f_n(x)=\frac{x^n}{\sqrt{3n}}$ for $x \in [0,1]$ is uniformly convergent [duplicate]
I'm completely confused by uniform convergence, but I put together the following proof just based on my other questions here and examples I read online. Discussion: Let $\epsilon \gt 0$ We want to ...
### Variant of dominated convergence theorem, does it follow that $\int f_n \to \int f$?
Suppose $f_n$, $g_n$, $f$ and $g$ are integrable, $f_n \to f$ almost everywhere, $g_n \to g$ almost everywhere, $|f_n| \le g_n$ for each $n$, and $\int g_n \to \int g$. Does it follow that \$\int f_n \...