How do I prove, that $\!C = \{f:\mathbb{R} \Rightarrow \mathbb{R}\mid f \space \text{ continuous}\}$ is equinumerous with $\mathbb{R}$?
12003
Cardinality of set of real continuous functions
131082
Proof that $\mathbb R/[0,1]$ and $\mathbb R$ are homeomorphic
46106
Prove that $D_r \circ D_s = \{ (x,y) \in \mathbb{R}^2 \ | \ |x-y|<r+s \}$, where $D_a = \{ (x,y) \in \mathbb{R}^2\ | \ |x-y| < a \}$
143798
Prove that, there exists no continuous function $f:\mathbb R\rightarrow\mathbb R$ with $f=\chi_{[0,1]}$ almost everywhere.
83326
$f:\mathbb R\to\mathbb R$ continuous function. Which of the following sets can not be image of $(0,1]$ under $f$?
9957
Prove if $f:[a,b] \rightarrow \mathbb{R}$ is integrable, then so is $|f|$ and $\left|\int_{a}^{b}f(x)dx\right| \leq \int_{a}^{b}|f(x)|dx$
92376
Let $f:A \rightarrow B$ and C ⊂A. Define f[C]={b∈B: b=f(a) for some a∈C}. Prove or disprove each of the statement
151803
Prove that $R$ is commutative
14495
I need to find all functions $f:\mathbb R \rightarrow \mathbb R$ which are continuous and satisfy $f(x+y)=f(x)+f(y)$
97104
Proof that continuous function on $ [0, 2]$ has $f(x) = f(x-1)$ for some $ x$
22869
Prove $[0, 1]$ and $(0, 1]$ are equinumerous by use of bijection
145077
A continuous bijection $f:\mathbb{R}\to \mathbb{R}$ is an homeomorphism?
61962
Show $R$ is right-Artinian but not left-Artinian
61640
Prove c exists for f:[0,1] -> R
157821
Let $S$ be a finite set. Show that any function $f: S \rightarrow \mathbb{R}^n$ is continuous on S. What if $S$ is countably infinite?
56567
Prove or disprove the uniform continuity of $f: \mathbb{C} \to \mathbb{C}, z \mapsto f(z)=\frac{z^2}{1+|z|}$
41872
Show that $\mathbb{R}^2$ is not homeomorphic to $\mathbb{R}^2 \setminus\{(0,0)\}$