Prove that if f is continuous at c and $f(c)0$ such that $\forall x \in Dom(f)$ and $x$ \in $(c- \delta , c + \delta)$...
13224
Continuous function positive at a point is positive in a neighborhood of that point
150235
$\epsilon$-$\delta$ proof that $f(x) = x \sin(1/x)$, $x \ne 0$, is continuous
100740
proof of that a continuous function has a fixed point
62118
Prove $\epsilon$-$\delta$ definition of continuity implies the open set definition for real function
76908
If $f'$ is differentiable at $a$ then $f'$ is continuous at $(a-\delta,a+\delta)$
140052
Show that $f(X - D) \cap f(D) = \varnothing$ with $f$ continuous in $X$, $D$ dense in $X$ and $f|_{D}$ homeomorphism
53497
Prove that the function is continuous and differentiable (not as easy as it sounds imo)
42084
Set of points of continuity are $G_{\delta}$
151410
Inequality of continuous functions
103495
Let $f$ be continuous. Then $\int_{0}^{x}\left[\int_{0}^{t}f\left(u\right)du\right]dt=\int_{0}^{x}f\left(u\right)\left(x-u\right)du$
69869
if $f$ is continuous on $\mathbb{R}$ and $f(r)=0,r \in \mathbb{Q}$, then $f(x)=0,x \in \mathbb{R}$
125754
Proving or disproving a statement about continuity and limits
20404
Prove that $f$ continuous and $\int_a^\infty |f(x)|\;dx$ finite imply $\lim\limits_{ x \to \infty } f(x)=0$
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 $a$
110146
I need to prove the continuity of $f(x)=\log x$ using a $\epsilon-\delta$ proof
77943
If $f$ is continuous, then there exists $x+\frac{1}{n}\in[0,1]$ and $f\left(x+\frac{1}{n}\right)=f(x)$
154559
For each $x\in X$ there exists a neighborhood of x s.t $A\cap U\neq \emptyset$ for only finitely many $A\in\mathcal{A}$. Show that f is continuous.