Prove that $f(x)=\begin{cases}1& \text{if x is rational},\\0 &\text{if x is irrational}\end{cases}$ is discontinuous at every real number.
43001
Proof for Dirichlet Function and discontinuous
26219
Suppose f is a function such that f'(x) = 1/x for all x > 0. Prove that if f(1) = 0, then f(xy) = f(x) + f(y) for all x,y > 0
7358
For each $y \in \mathbb{R}$ either no $x$ with $f(x) = y$ or two such values of $x$. Show that $f$ is discontinuous.
69707
If $f(x + y) \leq yf(x) + f(f(x))$ for all real numbers $x$ and $y$, then prove that $f(x) = 0$ for all $x ≤ 0$.
67020
Proving that for a smooth function if $f(\frac 1 k)=0 :\forall k\in \mathbb N$ then $f(x)=0 :\forall x\in[-1,1]$
147325
Proving that if $f(x)=x$ has no real solution then $f(f(x))=x$ has no real solution either
100765
If $f'(x) = 0$ for all $x \in \mathbb{Q}$, is $f$ constant?
41201
How prove there is no continuous functions $f:[0,1]\to \mathbb R$, such that $f(x)+f(x^2)=x$.
25467
Prove that $\lim f(x) =0$ and $\lim (f(2x)-f(x))/x =0$ imply $\lim f(x)/x =0$
120707
Proof regard to the function $ f\left(x\right)=\begin{cases} x^{2}\sin\left(\frac{1}{x^{2}}\right) & x\neq0\\ 0 & x=0 \end{cases} $
81223
Real Analysis Question: continuity of $f(x) = \sum_{q_n<x} 1/n^2$
150346
If $f$ is an increasing function defined on $[a,b]$ . Then, $f$ has countable number of discontinuities
118873
Prove that $f(x)=0$ for all $x\in\mathbb{R}$.
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 $\mathbb Q$.
72694
Proving that $f(x) = g(x)$ for some $x \in [a,b]$ if $f,g$ continuous, $f(a)  g(b)$
120935
Why is every continuous function on the reals determined by its value on rationals?
87131
Discontinuous at rationals and differentiable at irrationals?