query: For a real valued function $f(x,y)$ on $\mathbb{R}^2$ which is in $L_2$, show that $f(x+ε,y+ε) → f(x,y)$ in $L_2$ when $ε → 0.$
positive_passages.docid: 142025
positive_passages.text: How can I show that if $f\in L^p(a, b)$ then $\lim_{t\to 0^{+}}\int_{a}^b |f(x+t)-f(x)|^p\ dx=0$..
negative_passages.docid: 20273
negative_passages.text: Prove that if $|f(x)| \leq x^2$, then the function is continuous and differentiable at $x=0$.
negative_passages.docid: 61500
negative_passages.text: If f is continous and exist real numbers show that $f(x)=0$
negative_passages.docid: 116548
negative_passages.text: How to prove the real value function $f$ is convex on $[a,d]$ if $f$ is convex on both $[a,c]$ and $[b,d]$? ($a<b<c<d$).
negative_passages.docid: 149564
negative_passages.text: Showing that a function is in $L^1$
negative_passages.docid: 98772
negative_passages.text: Continuity of a function in $\mathbb{R}$
negative_passages.docid: 153730
negative_passages.text: Show that $f(x)=x^2$ is continuous at $a=2$ using the $\delta-\epsilon$ definition of continuity.
negative_passages.docid: 76485
negative_passages.text: $f_n(x) \to f(x)$ and $\int |f_n|^2 \, d\mu \to \int |f|^2 \, d\mu$. Use Egorov's theorem to show $f_n \to f$ in $L^2(d\mu)$.
negative_passages.docid: 69707
negative_passages.text: If $f(x + y) \leq yf(x) + f(f(x))$ for all real numbers $x$ and $y$, then prove that $f(x) = 0$ for all $x ≤ 0$.
negative_passages.docid: 141223
negative_passages.text: $f(x^2) = 2f(x)$ and $f(x)$ continuous
negative_passages.docid: 144589
negative_passages.text: Proving that if $f(x) \le g(x)$ then $f(x_0) \le g(x_0)$
negative_passages.docid: 68409
negative_passages.text: Consider $f(x,y)=\sup_{θ,φ}\{||e^{i θ }x+ e^{i φ }y||^2: θ,φ∈R\}$ in which $x,y\in\mathbb C^n$
negative_passages.docid: 19916
negative_passages.text: $f(x)=f(x^2+ 1/4)$ , $f$ is continuous from $\mathbb{R}$ to $\mathbb{R}$
negative_passages.docid: 155232
negative_passages.text: Prove that inequality is true for $x>0$: $(e^x-1)\ln(1+x) > x^2$
negative_passages.docid: 126821
negative_passages.text: Let $f:[0,1]\to \mathbb{R}$. Show that $\{a \in [0,1]: f(a)\neq 0\} = \bigcup_{n=1}^{\infty} \{a \in [0,1]: |f(a)|\geq 1/n\}$
negative_passages.docid: 76148
negative_passages.text: Show that $\exists c \in \mathbb{R} : f(c) = 0$
negative_passages.docid: 85092
negative_passages.text: Proving $ f(x) = x^2 $ is not uniformly continuous on the real line