How does one compute the cardinality of the set of functions $f:\mathbb{R} \to \mathbb{R}$ (not necessarily continuous)?

Could somebody explain to me how to prove that the cardinality of all real continuous functions is $c$ ? The first problem is that I don't know how to show that each real continuous function $f: X \rightarrow Y$ is uniquely determined by its values for $x \in Q $. Secondly, how to show that $R^...

The set of all $\mathbb{R\to R}$ continuous functions is $\mathfrak c$. How to show that? Is there any bijection between $\mathbb R^n$ and the set of continuous functions?

Trigonometric limits: with sine, at $0$ $\displaystyle \lim\limits_{x\to 0}\frac{\sin x}x=1$ 1, 2, 3, 4, 5, 6, 7. $\displaystyle \lim_{x \to 0} (1+ \sin 2x)^{\frac{1}{x}}$ $\displaystyle \lim_{(x,y)\to(0,0)}\frac{(\sin^2x)(e^y-1)}{x^2+3y^2}$ $\displaystyle \lim_{x \to 0} \sin(x)$ $\displaystyle...

Lately, I've been very confused about the weird properties of limits. For example, I was very surprised to find out that $\lim_{n \to \infty} (3^n+4^n)^{\large \frac 1n}=4$ , because if you treat this as an equation, you can raise both sides to the $n$ power, subtract, and reach the wrong conclus...

I can't understand why should $\sin(0)$ exist, because if an angle is $0^{\circ}$, then the triangle doesn't exist i.e. there is no perpendicular or hypotenuse. However, if we take $\lim_{x \to 0} \sin(x)$, then I can understand $$\lim_{x \to 0} \sin(x) = 0$$ since perpendicular $\approx$ 0. So a...

Trigonometric limits: with sine, at $\infty$ $\displaystyle \lim_{n \rightarrow \infty}\frac{\sin 1+2\sin \frac{1}{2}+\cdots+n\sin \frac{1}{n}}{n}$ 1, 2 $\displaystyle \displaystyle\lim_{x \to +\infty} x \sin x$ $\displaystyle \lim \limits_{n \to \infty} |\sin(\pi \sqrt{n^2+n+1})|$ $\displaysty...