Is the function $f(x)= \begin{cases} 0, & \text{if}\ x\ \text{is rational} \\ x, & \text{otherwise} \end{cases}$, continuous?
98666
Continuity defined for rational and irrational numbers
158309
Find a function that is continuous only on rationals
94109
$f$ a real, continuous function, is it measurable?
86472
$f:\mathbb R \to \mathbb R^n$ be such that $G(f):=\{(x,f(x)):x \in \mathbb R\}$ is closed and connected in $\mathbb R^{n+1}$ , is $f$ continuous ?
139744
For a continuous function $f$ show $\exists c\in (0,1)$ s.t $f(c)=3c^2$
69869
if $f$ is continuous on $\mathbb{R}$ and $f(r)=0,r \in \mathbb{Q}$, then $f(x)=0,x \in \mathbb{R}$
104369
If $f$ continuous and $f(x^2) = f(x)$, then $f$ is a const
54940
If integral is zero and function is continuous and non negative, then what about the function?
93021
$f:X\longrightarrow\mathbb{R}$ is continuous iff $\{x\in X:f(x)\geq\alpha\}$ and $\{x\in X:f(x)\leq\alpha\}$ are closed $\forall\alpha\in\mathbb{R}$
33499
For which $a\in \mathbb{R}$ is $f(x,y) = \frac{xy^a}{x^2+y^2}$ with $f(0,0)=0$ continuous?
100187
Show function is continuous at origin
88640
Is $\sqrt x$ continuous at $0$? Because it is not defined to the left of $0$
40209
$f : S^1 \to\mathbb R$ is continuous then $f(x)=f(-x)$ for some $x\in S^1$
145919
If $f: \Bbb{R}\rightarrow\Bbb{R}$ is continuous at $0$ and $f(x)=f(2x)$ for each $x\in\Bbb{R}$ then $f$ is constant.
158888
Prove that the function $f(x)=\frac{x}{x^2+1}$ is continuous at $x=-1$.
122160
$f:\mathbb R \to \mathbb R$ be a continuous such that $f(f(f(x)))=x,\forall x \in \mathbb R$ , is $f(x)=x , \forall x \in \mathbb R$?
154521
Linear functional $f$ is continuous at $x_0=0$ if and only if $f$ is continuous $\forall x\in X$?