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 Jun 6 answered Function defined by an integral. Jun 5 revised (Ito lemma proof): convergence of $\sum_{i=0}^{n-1}f(W(t_{i}))(W(t_{i+1})-W(t_{i}))^{2}.$ added 3 characters in body Jun 5 revised (Ito lemma proof): convergence of $\sum_{i=0}^{n-1}f(W(t_{i}))(W(t_{i+1})-W(t_{i}))^{2}.$ added 194 characters in body Jun 5 revised (Ito lemma proof): convergence of $\sum_{i=0}^{n-1}f(W(t_{i}))(W(t_{i+1})-W(t_{i}))^{2}.$ added 194 characters in body Jun 5 asked (Ito lemma proof): convergence of $\sum_{i=0}^{n-1}f(W(t_{i}))(W(t_{i+1})-W(t_{i}))^{2}.$ May 29 comment If $f$ and $g$ are integrable functions on $I=[a,b]$ and if $h(x):=\inf(f(x), g(x))$ for all $x \in I$, prove that $h$ is integrable at $I$. $P=\{x_{0},x_{1},\ldots,x_{n}\}$. It could be that, say, $P_{1}=\{x_{0},x_{3},x_{5},x_{11},\ldots,x_{n-1}\}$ and $P_{2}=\{x_{1},x_{2},x_{4},\ldots,x_{n}\}$. Then $P_{1}$ and $P_{2}$ are partitions of $I$ and they are sub-partitions of $P$ in the sense that $P=P_{1}\cup P_{2}$. The index sets in this example are $I_{1}=\{0,3,5,11,\ldots,n-1\}$ and $I_{2}=\{1,2,4,\ldots,n\}.$ The manner in which the sub-partitions are obtained depend on $f$ and $g$ in accordance to the construction in the solution above. Perhaps the notation of $I_{j}$ is confusing-it has nothing to do with the interval $I$ May 29 revised If $f$ and $g$ are integrable functions on $I=[a,b]$ and if $h(x):=\inf(f(x), g(x))$ for all $x \in I$, prove that $h$ is integrable at $I$. added 810 characters in body May 29 revised If $f$ and $g$ are integrable functions on $I=[a,b]$ and if $h(x):=\inf(f(x), g(x))$ for all $x \in I$, prove that $h$ is integrable at $I$. added 810 characters in body May 29 revised If $f$ and $g$ are integrable functions on $I=[a,b]$ and if $h(x):=\inf(f(x), g(x))$ for all $x \in I$, prove that $h$ is integrable at $I$. added 810 characters in body May 29 answered If $f$ and $g$ are integrable functions on $I=[a,b]$ and if $h(x):=\inf(f(x), g(x))$ for all $x \in I$, prove that $h$ is integrable at $I$. May 14 comment Bound on uniform norm of convolution of $L^p$ functions Oh. I didn't even know that notation was still used! May 13 comment Bound on uniform norm of convolution of $L^p$ functions Don't you need $p^{-1}+q^{-1}=u^{-1}+1$? May 13 answered Diffuse equation-type PDE: Help me! May 12 comment How do I convert the limit definition of differentiability to different variables? Because $h$ is just a displacement vector (from the point in question, $(x_{0},y_{0})$), and you send it to $0$. Being a vector, it has coordinates. Call them $(x,y)$, or anything you wish. May 12 revised How do I convert the limit definition of differentiability to different variables? added 236 characters in body May 12 answered How do I convert the limit definition of differentiability to different variables? May 12 revised A calculus problem with functions such that $f''(x) = g(x)$ and $g''(x) = f(x)$ added 40 characters in body May 12 revised A calculus problem with functions such that $f''(x) = g(x)$ and $g''(x) = f(x)$ added 195 characters in body May 12 answered A calculus problem with functions such that $f''(x) = g(x)$ and $g''(x) = f(x)$ May 12 answered Examine the convergence of $\left({\cos n\over n}\right)$