$f$ is continuous, $\forall x , \lim_{n \to \infty}f(nx)=0 \implies \lim_{x \to \infty}f(x)=0$
36248
A classical problem about limit of continuous function at infinity and its connection with Baire Category Theorem
144013
If $f$ is differentiable at $x = x_0$ then $f$ is continuous at $x = x_0$.
66441
Prove that $|x|<1$ implies that $\lim_{n \to \infty} x^n=0$
125754
Proving or disproving a statement about continuity and limits
92089
If $f:[a,b]\to \mathbb{R}$ is continuous and nonnegative and $\int_a^b{f}=0$, then $f(x)=0$ for all $x\in [a,b]$
40297
Proof of uniform limit of Continuous Functions
153267
Proving $\lim\limits_{x \to \infty} xf(x)=0$ if $\int_{0}^{\infty}f(x) dx$ converges.
51594
$\lim_{n→∞} f(n+x)=0$ results in $ \lim_{x→∞} f(x)=0$
9614
How does the existence of a limit imply that a function is uniformly continuous
25467
Prove that $\lim f(x) =0$ and $\lim (f(2x)-f(x))/x =0$ imply $\lim f(x)/x =0$
46101
If $f(x)$ is continuous on $[a,b]$ and $M=\max \; |f(x)|$, is $M=\lim \limits_{n\to\infty} \left(\int_a^b|f(x)|^n\,\mathrm dx\right)^{1/n}$?
154521
Linear functional $f$ is continuous at $x_0=0$ if and only if $f$ is continuous $\forall x\in X$?
10428
Proving that $\lim\limits_{x\to\infty}f'(x) = 0$ when $\lim\limits_{x\to\infty}f(x)$ and $\lim\limits_{x\to\infty}f'(x)$ exist
105652
Suppose $f: \mathbb{R} \to \mathbb{R}$ is function such that $f^3$ is continuous on $\mathbb{R}$. Prove that $f$ is continuous on $\mathbb{R}$
94737
How to show $f$ is continuous at $x$ IFF for any sequence ${x_n}$ in $X$ converging to $x$ the sequence $f(x_n)$ converges in $Y$ to $f(x)$
20404
Prove that $f$ continuous and $\int_a^\infty |f(x)|\;dx$ finite imply $\lim\limits_{ x \to \infty } f(x)=0$
132013
How do I show that if $f : [0,1] \to [0,1]$ is continuous, then there is some $c$ such that $f(c) = c$?