Prove or Disprove that if $f_x(x_0,y_0)$ and $f_y(x_0,y_0)$ both exist, then f is continuous at $(x_0,y_0)$
11017
Does existence of partial derivatives implies continuity at a point $(x_0,y_0)$?
46629
Prove if $f(x) = g(x)$ for each rational number x and $f$ and $g$ are continuous, then $f = g$
129551
Does $\lim_{h\rightarrow 0}\ [f(x+h)-f(x-h)]=0$ imply that $f$ is continuous?
47381
If $\lim_{x\to\infty} [f(x+1)-f(x)] =l$ then $\lim_{ x\to\infty}f(x)/x =l$ ($f$ is continuous)
74950
Show a function for which $f(x + y) = f(x) + f(y) $ is continuous at zero if and only if it is continuous on $\mathbb R$
34940
Continuity of $f(x) = x^n$
149095
$f(\mathbb{R}\setminus \mathbb{Q}) \subseteq \mathbb{Q}$ and $f(\mathbb{Q}) \subseteq \mathbb{R}\setminus \mathbb{Q}$ imply that $f$ is not continuous
153730
Show that $f(x)=x^2$ is continuous at $a=2$ using the $\delta-\epsilon$ definition of continuity.
110458
Show that if $f$ is continuous on $(0,1)$ and $\lim_{x \to 0}f(x) = + \infty$ then $f$ is not uniformly continuous
85623
Proof of continuity implied by differentiability
22303
How do I prove that $f(x)f(y)=f(x+y)$ implies that $f(x)=e^{cx}$, assuming f is continuous and not zero?
16735
If $f,g:X \to Y$ are continuous and $Y$ is $T_2$, then $\{x \in X\,|\,f(x)=g(x)\}$ is closed
80643
Show that f(x) defined on [0,infinity) R to is continuous at 0.
155923
continuity of $f(x,y)$ in $\mathbb R^2$.
118207
Prove that the product of two continuous functions is continuous
136011
Must $f$ be continuous?
100740
proof of that a continuous function has a fixed point