Which values of $\alpha, \beta > 0$ make $f(x,y)=\frac{|{x}|^{\alpha}|{y}|^{\beta}}{x^2 + y^2}$ continuous?
13249
Multivariable limit proof: $\lim\limits_{(x,y)\rightarrow (0,0)}\frac{\left|x\right|^a\left|y\right|^b}{\left|x\right|^c + \left|y\right|^d} = 0$
20273
Prove that if $|f(x)| \leq x^2$, then the function is continuous and differentiable at $x=0$.
24017
Let $f: \Bbb R \to [0,\infty)$ be a continuous function such that $g(x)=(f(x))^2$ is uniformly continuous. Which of the following is always true?
143174
property of continuous function
81223
Real Analysis Question: continuity of $f(x) = \sum_{q_n<x} 1/n^2$
40576
Intermediate value property problem and continuous function
134752
Proof (epsilon delta) for the continuity of a function at a point
129011
Is there limit $ \lim_{(x,y) \to (0,0)} \frac{x^3}{x^2 + y^2}$?
145017
Example of a function $f$ which is nowhere continuous but $|f|$ should be continuous at all points
37489
Show that $ \lim_{(x,y) \to 0} \frac {|x|^ \alpha |y|^ \beta} {|x|^ \gamma + |y|^ \delta} \text {exists} \iff \alpha/\gamma + \beta/\delta > 1.$
13510
Multivariable Delta Epsilon Proof $\lim_{(x,y)\to(0,0)}\frac{x^3y^2}{x^4+y^4}$ --- looking for a hint
77943
If $f$ is continuous, then there exists $x+\frac{1}{n}\in[0,1]$ and $f\left(x+\frac{1}{n}\right)=f(x)$
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.
150235
$\epsilon$-$\delta$ proof that $f(x) = x \sin(1/x)$, $x \ne 0$, is continuous
141945
Why is $f(x) = \sqrt{x}$ is continuous at $x=0$? Does $\lim_{x \to 0^{-}} \sqrt{x}$ exist?
5419
Continuous functions with $f^2(x)=g^2(x)\neq 0$
39179
Why is $(0, 0)$ not a minimum of $f(x, y) = (y-3x^2)(y-x^2)$?