7
votes
Question on countability of set of all functions $f:\mathbb{N}\to\mathbb{N}$
L2: Define the function $g:F\to\mathbb{N}$ given by: $g(f)=2^{f(1)}3^{f(2)}5^{f(3)}\dots p_{i}^{f(i)}\dots$ where $p_{i}$ is the i-th prime.
It already fails here. Say $f(x)=1$. Then what is $g(f)$ ...
- 46.3k
6
votes
A functional equation in Olympiad Mathematics
Notice that $f\equiv 0$ is a solution, so it need not be bijective. Suppose we have another solution with $f\not\equiv0$ and let $x_0$ be such that $f(x_0)\neq0$. Then, plugging $y=x_0$ we get
$$f(xf(...
- 1,386
4
votes
Why is a function not differentiable at the end-points of its domain?
Is there a source telling you that you can't say $x^2$ on $[0,8]$ is differentiable at the endpoints?
The way you pose the question makes me suspect you are studying the Mean Value Theorem, which is ...
- 50.8k
3
votes
Accepted
Prove that $\sqrt{x} > \ln(x)$ holds true for all $x > 0$.
Geometric proof for $e^x>2x$:
the tangent line of $e^x$ at the point $(\ln2,2)$ is
$$y=2x+2(1-\ln2).$$
Because $e^x$ is strictly convex on $\mathbb R$, so
$$e^x\geq2x+2(1-\ln2)>2x,\quad \...
- 10.3k
2
votes
Question on countability of set of all functions $f:\mathbb{N}\to\mathbb{N}$
You are correct in that there is a catastrophic error is in I.2, in fact $g(f)$ as constructed is not finite even given that $f$ is bounded.
RE I.3: You do not need $g$ to be surjective. More ...
- 22k
2
votes
Accepted
Is it implied in the question that $x$ is greater than $-1$?
Graphically, it is clear that $$f(x)=\begin{cases}x+1\,\forall\, x\le0\\ \sqrt{1-x}\,\forall\, x\in[0,1]\end{cases}$$
The $x$-axis intersects the graph of $f(x)$ only at $(-1,0)$ and $(1,0)$, implying ...
- 4,468
2
votes
Prove that $\sqrt{x} > \ln(x)$ holds true for all $x > 0$.
In what follows, I will assume you are familiar with elementary calculus.
Consider the function $f(x) = \sqrt{x} - \ln(x)$. Differentiating it, we get $f'(x) = \frac{1}{2\sqrt{x}} - \frac{1}{x}$. Thus,...
- 2,807
2
votes
Prove that $\sqrt{x} > \ln(x)$ holds true for all $x > 0$.
Substituting $x=e^{2t}$, the given inequality is equivalent to
$$
e^t \ge 2t,
$$
for all real $t$. Consider the function $f(t) = e^t -2t$. It’s a sum of two convex functions, so it is convex. In fact, ...
- 3,658
2
votes
A functional equation in Olympiad Mathematics
First note that the zero function is a solution. Let now $f$ be another solution. Then, $(x+1)f(y)$ can take any real value hence $f$ is surjective, so that the functional equation is equivalent to
$$\...
- 44.2k
1
vote
Range of $y$ when $y = \frac{1}{x^2}$ for negative values of $x$
First of all, the given domain should be $(-6,0)\cup(0,6)$ as $\frac1{x^2}$ is not defined at $x=0$.
Note that for finding range of a function, the graphical approach is much quicker and easily helps ...
- 4,468
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