To prove $f(x)=f(h(x))$ for homeomorphism $h$ and continuous $f$.
134703
Let $X$ be a connected space, $f:X\to X$ continuous involution, $g:X\to\mathbb{R}$ continuous. Prove that $\exists$ $x\in X$ such that $g(x)=g(f(x))$.
111756
How do I prove that then $f(x) = 1$ for some $x ∈ [0, 1]$?
104949
How would I prove that this function is affine if $f(x+h)-f(x)=hf'(x)$?
157597
Homeomorphism between $\mathbb{Q}$ and $\mathbb{Q}(>0)$, and $\mathbb{Q}(\ge 0)$
57023
If $f,g$ are continuous at $a$, show that $h(x)=\max\{f(x),g(x)\}$ and $k(x)=\min\{f(x),g(x)\}$ are also continuous at $a$
81488
If a continuous function$f: (X, \mathscr T_X) \to (Y, \mathscr T_Y)$ is injective (Given $Y$ is Hausdorff), show that X is hausdorff
104369
If $f$ continuous and $f(x^2) = f(x)$, then $f$ is a const
6825
If $f: \mathbb Q\to \mathbb Q$ is a homomorphism, prove that $f(x)=0$ for all $x\in\mathbb Q$ or $f(x)=x$ for all $x$ in $\mathbb Q$.
113975
Let $f:\mathbb{R}\to\mathbb{R}$, $f(kx)=f(x)$. Show that if $f$ is continuous at $0$ then $f$ is a Constant function
2072
Demonstrate that if $f$ is surjective then $X = f(f^{-1}(X))$
28399
Showing $f$ is constant
37966
If $f$ is continuous and $f(x+y) = f(x)+f(y)$, then $f(x) = cx$ for all $x \in \mathbb{R}$
21391
Proving Functions are Surjective
66391
Show that $f(x+y)=f(x)+f(y)$ implies $f$ continuous $\Leftrightarrow$ $f$ measurable
72694
Proving that $f(x) = g(x)$ for some $x \in [a,b]$ if $f,g$ continuous, $f(a)  g(b)$
29276
technique of proving continuity complex function
150554
proof that $f'$ is continuous at $a$