Show that there exist a continuous function $f:X\to [0,1]$ such that $f^{-1}(\{0\})=A$ and $f(B)=\{1\}$.
12349
a construction of a continuous function from a normal space to [0,1]
158940
Give an example of a continuous function $f : [0, ∞) \mapsto [0, ∞)$ such that $\int_{0}^{\infty}f(x)dx$ exists but $f$ is unbounded.
90565
If $f:[0,1] \rightarrow \mathbb{R}$ is continuous function such that $f(0)=f(1)$ then there exists $ x\in [0,1]$ such that $f(x) = f(x+\frac{1}{n})$.
141223
$f(x^2) = 2f(x)$ and $f(x)$ continuous
76148
Show that $\exists c \in \mathbb{R} : f(c) = 0$
2072
Demonstrate that if $f$ is surjective then $X = f(f^{-1}(X))$
97773
If $f$ is continuous with $f(x) = f(2x),f(1) = 3$, then what is $ \int_{-1}^{1}f(f(x))\,dx$?
49592
Show that, if $f:A\to B$ is a function, with $A$ and $B$ being finite sets, and $|A|=|B|$, then $f$ is one to one iff $f$ is onto.
5419
Continuous functions with $f^2(x)=g^2(x)\neq 0$
13575
Need help for proving that: $f(f^{-1}(A)) ⊆ A$
95469
Suppose $f:[0,1] \Rightarrow \mathbb{R}$ is continuous and $\int_0^x f(x)dx = \int_x^1 f(x)dx$. Prove that $f(x) = 0$ for all $x$
76610
Show that $f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B)$
90619
For closed subsets $A,B \subseteq X$ with $X = A \cup B$ show that $f \colon X \to Y$ is continuous iff $f|_A$ and $f|_B$ are continuous.
68165
Find a function $f: \mathbb{R} \to \mathbb{R}$ that is continuous at precisely one point?
54797
Find the continuous function defined over $[0,1]$ such that $\int_0^1f(x){\rm d}x=1, \int_0^1 xf(x){\rm d}x=a$ and $\int_0^1x^2f(x){\rm d}x=a^2 .$
154034
Suppose that $f(x)$ is continuous on $(0, \infty)$ such that for all $x > 0$,$f(x^2) = f(x)$. Prove that $f$ is a constant function.
113503
Show that exists a not decreasing function that $f:(a,b)\rightarrow \mathbb{R}$ that is continuous only in $(a,b)\setminus D$.