query: Let $~f~$ be a non-negative continuous function on $~\mathbb{R}~$ such that $~\int_{0}^{∞} f(x)~dx~$ is finite,then $~\lim_{x\to ∞} f(x) = 0~$.
positive_passages.docid: 20404
positive_passages.text: Prove that $f$ continuous and $\int_a^\infty |f(x)|\;dx$ finite imply $\lim\limits_{ x \to \infty } f(x)=0$
negative_passages.docid: 16212
negative_passages.text: If $f$ is continuous, nonnegative on $[a, b]$, show that $\int_{a}^{b} f(x) d(x) = 0$ iff $f(x) = 0$
negative_passages.docid: 29180
negative_passages.text: If $f(x/n)\to0$ when $n\to\infty$, for every $x$ and $f$ is continuous, then $f(x)\to0$ when $x\to0$, or not?
negative_passages.docid: 40915
negative_passages.text: Proof if $f$ continuous in $x_0$ then there is a neighbourhood of $x_0$ so f bounded.
negative_passages.docid: 21301
negative_passages.text: If nonnegative $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$, then $\int_0^1 \Big| \frac{f''(x)}{f(x)} \Big| \,dx >4$
negative_passages.docid: 94971
negative_passages.text: For every $\epsilon>0$ there exists $\delta>0$ such that $\int_A|f(x)|\mu(dx) < \epsilon$ whenever $\mu(A) < \delta$
negative_passages.docid: 58361
negative_passages.text: Is there a continuous function such that $\int_0^{\infty} f(x)dx$ converges, yet $\lim_{x\rightarrow \infty}f(x) \ne 0$?
negative_passages.docid: 46101
negative_passages.text: If $f(x)$ is continuous on $[a,b]$ and $M=\max \; |f(x)|$, is $M=\lim \limits_{n\to\infty} \left(\int_a^b|f(x)|^n\,\mathrm dx\right)^{1/n}$?
negative_passages.docid: 103495
negative_passages.text: Let $f$ be continuous. Then $\int_{0}^{x}\left[\int_{0}^{t}f\left(u\right)du\right]dt=\int_{0}^{x}f\left(u\right)\left(x-u\right)du$
negative_passages.docid: 58597
negative_passages.text: A characterization of functions from $\mathbb R^n$ to $\mathbb R^m$ which are continuous
negative_passages.docid: 30940
negative_passages.text: Prove the integral of $f$ is positive if $f ≥ 0$, $f$ continuous at $x_0$ and $f(x_0)>0$
negative_passages.docid: 127868
negative_passages.text: If $f$ is nonnegative and integrable then $F(x) = \int_{-\infty}^x f$ is continuous.
negative_passages.docid: 136594
negative_passages.text: Proving the existence of a sequence of polynomials convergent to a continuous function $f$.
negative_passages.docid: 131992
negative_passages.text: A problem on continuous functions
negative_passages.docid: 142637
negative_passages.text: If $f \in \mathcal{L}^1(\mathbb{R})$ then $\lim_{h\rightarrow 0}\int_{-\infty}^\infty |f(x+h)-f(x)|dx\rightarrow 0$
negative_passages.docid: 95469
negative_passages.text: Suppose $f:[0,1] \Rightarrow \mathbb{R}$ is continuous and $\int_0^x f(x)dx = \int_x^1 f(x)dx$. Prove that $f(x) = 0$ for all $x$
negative_passages.docid: 52632
negative_passages.text: $f\colon(0,\infty)\to \mathbb R$ be continuous ; $f(x)\le f(nx) , \forall n \in \mathbb N , \forall x >0$ , then $\lim_{x\to \infty} f(x)$ exists?