Prove or give a counterexample: If $f,g$ are continuous and proper, then the product $fg$ is proper.
111246
product of two continuous proper maps
105233
Is there a counterexample for the claim: if $A \oplus B\cong A\oplus C$ then $B\cong C$?
90816
Why does the product topology allow proper subsets for only finitely many elements?
144614
If $f$ is continuous on $[a,b]$ and $F(x) = \sup f([a,x])$. Prove that $F$ is continuous on $[a,b]$ .
124444
If $f$ s discontinuous at $x_0$ but $g$ is continuous there, then $f+g$?
37966
If $f$ is continuous and $f(x+y) = f(x)+f(y)$, then $f(x) = cx$ for all $x \in \mathbb{R}$
85612
Let functions $ f: A \to B$ and $g: B \to A$, suppose that $ g \circ f = i_A$. Prove that if $f$ is onto, then $g$ is one-to-one
40609
Prove that $f(X)$ and $g(Y)$ are independent if $X$ and $Y$ are independent
74364
Show that function $f$ has a continuous extension to $[a,b]$ iff $f$ is uniformly continuous on $(a,b)$
11974
product of two non zero continuous function is zero
118332
proof $f(\overline{A})\subseteq \overline{f(A)} \Leftrightarrow f$ continuous
94571
How can we prove that $N_G(H)=H$ implies $N_{f(G)}(f(H))=f(H)$
46629
Prove if $f(x) = g(x)$ for each rational number x and $f$ and $g$ are continuous, then $f = g$
97104
Proof that continuous function on $ [0, 2]$ has $f(x) = f(x-1)$ for some $ x$
28730
product of two uniformly continuous functions is uniformly continuous
105455
Prove that "injective function $f:X\to Y$ exists" and "surjective function $g:Y\to X$ exists" is logically equivalent.
118207
Prove that the product of two continuous functions is continuous