Intereting Posts

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Borel Measures: Atoms vs. Point Masses
On Constructions by Marked Straightedge and Compass
Common misconceptions about math
In what conditions every ideal is an extension ideal?
Prerequisite for Petersen's Riemannian Geometry
Total ordering on complex numbers
Good book for self study of a First Course in Real Analysis

The question asks to prove that there exists a unique function defined on $\Bbb R$ and satisfying the following conditions:

1) $f(1) = a$ $(a>0, a \neq 0)$

2) $f(x_1) \cdot f(x_2) = f(x_1 + x_2)$

- Uniform convergence of the Bergman kernel's orthonormal basis representation on compact subsets
- How to show $\lim_{n \to \infty} a_n = \frac{ + + + \dotsb + }{n^2} = x/2$?
- How to prove that $\,\,f\equiv 0,$ without using Weierstrass theorem?
- Does a dense $G_\delta$ subset of a complete metric space without isolated points contain a perfect set?
- If $f(x + y) = f(x) + f(y)$ showing that $f(cx) = cf(x)$ holds for rational $c$
- the elements of Cantor's discontinuum

3) $f(x) \rightarrow f(x_0)$ as $x \rightarrow x_0 $

Since the text goes into detail constructing the exponential function and proving these properties, I assume I only have to show uniqueness. i.e. showing that if functions $f$ and $g$ satisfy these properties, then $f=g$.

$f(1) = g(1)$ by property (1).

Then for $n\in \Bbb N$, $(f-g)(n)$ $=$$f(n) – g(n)$ $= f(1)^n -g(1)^n $ by property (2) and induction.

But $f(1) = g(1) = a > 0$ , which implies $f(1)^n -g(1)^n = 0$ and hence $f(n) =g(n)$

We have $f(1)-g(1)$ $=$ $f(\frac1n \cdot n) -g(\frac1n \cdot n)$ $=$$f(\frac1n)^n-g(\frac1n)^n = 0$.

Now $f(x) = f(\frac x2)^2 \ge 0$, so $f(\frac1n) =g(\frac1n)$.

$f(\frac mn ) -g (\frac mn ) $ $=$ $f(\frac1n)^m – g(\frac1n)^m$ which implies that $f(\frac mn) – g(\frac mn)$ $=0$.

For $x \in \Bbb R$, $\lim \limits_{\Bbb Q \in r \to x}$ $(f-g)(r) =0$: for any $\epsilon > 0$, we can choose a $\delta > 0$ such that $|r – x| < \delta $ and $|f(r) – g(r)| = 0 < \epsilon $.

By property 3 we also have $\lim \limits_{\Bbb Q \in r \to x}(f-g)(r) =\lim \limits_{\Bbb Q \in r \to x} f(r) – \lim \limits_{\Bbb Q \in r \to x} g(r) = f(x) – g(x)$.

Thus $f(x) = g(x)$ $\forall x \in \Bbb R$.

Is this approach correct?

A similar question is asked for the logarithmic function.

- How to show $\lim_{n \to \infty} a_n = \frac{ + + + \dotsb + }{n^2} = x/2$?
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- Questions about $f: \mathbb{R} \rightarrow \mathbb{R}$ with bounded derivative
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- What's the relationship between a measure space and a metric space?
- functions with eventually-constant first difference
- From $e^n$ to $e^x$
- On distributions over $\mathbb R$ whose derivatives vanishes
- existence of a special function
- Accumulation points of the set $S=\{(\frac {1} {n}, \frac {1} {m}) \space m, n \in \mathbb N\}$

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