real-analysis

  1. convergence of a nonnegative monotonic sequence

  2. Its possible integrate this with elementary calculus?
  3. If a function f: R-> R is monotone increasing on R and f(R) is compact, is f continuous?

  4. Is the space $(X,p)$ complete? If not what is its completion?

  5. Equivalent Sequential Definitions of Continuity

  6. Arzela Ascoli counterexamples

  7. A function $f$ such that $\lim_{x\to b}f(x)=+\infty$ and $\lim_{x\to b}f'(x)=-\infty$

  8. Showing that replacing functions with values preserves concavity

  9. Convergence of an improper integral $ \log( 1 + 2\operatorname{sech}x)$

  10. If we assume merely that the partials $D_jf$ exist in a neighborhood of $a$ and are continuous at $a$ then $f$ is differentiable in $a$.

  11. Show that either there exists an $N$ so that $r_n=r$ for $n \ge N,$ or the set of numbers $\{q_n\}$ is unbounded.

  12. convergent or divergent $\int_{-4}^{1} \frac{dz}{(z + 3)^3}$
  13. If $a_n \ge 0$ and $a_n \to 0$, then $\sum a_n$ converges
  14. Does $f'(x)>0$ a.e. imply that $f$ is strictly monotone?

  15. Find all $\alpha$ for which $T_\alpha$ is well-defined, injective, $J o T_\alpha $ is a contraction, $T_\alpha$ is a homeomorphism.

  16. Covariance function $K(s,t)=\exp(-\vert t-s\vert).$

  17. Improper integral for $x^{-a} \sin(x)$ for $1<a<2$

  18. Comparative status of $\sum a_n$ and $\sum {a_n}^{[2]}$.
  19. Prove $K$ is an approximation of identity
  20. How to find $\sum \left(1 + 1/2 + \dots + 1/(n + k + 1)\right)\frac{1}{n(n + 1)...(n + k + 1)}$?
  21. Given $K$ an ordered field, if, for every $a\in K$ there's a sequence of rationals $(a_{n})$ with limit $a$, then K is archimedean

  22. Fubini in $\mathbb{R^2}$

  23. Examples of Axiom of Choice used in introductory-level undergradute math

  24. An inequality in the proof of $L(p,q_1)\subset L(p,q_2)$ in Stein-Weiss
  25. Suppose $f$ is differentiable on $[a,b]$ and $\lim_{x\to a+}f'(x)=C$ then determine the right-hand derivative at $a$

  26. what is the value of $\sum \frac{x^n}{1+x^n}$ on $x\in [0,+1)$
  27. How to prove this equivalent
  28. Showing Convergence in a Set of Sequences

  29. General partial fraction decomposition

  30. Find all directions such that they decrease the function value after taking a small step towards that direction
  31. Limit involving integral in $\mathbb{R}^n$

  32. Derivative of gradient

  33. If $f(x)+g(x)$ is strictly concave, then is $x+(g \circ f^{-1})(x)$ also strictly concave?

  34. How irregular can $f'$ be beyond Darboux's Theorem?

  35. Abel Differential equation of second kind

  36. Is convergent $\int_0^{+\infty}\sin\log\left(\frac{f(x)}{xf'(x)}\right)dx$, if $f(x)$ is of slow increase in spirit of Jakimczuk's definition?

  37. Upper hemicontinuity and closed graphs

  38. Bound for a C2 function using Mean Value Theorem
  39. Matrix-valued function that implies an upper bound

  40. Can we always choose a countable dense subset of an open set of the complex plane?

  41. Prove that $\lim_{x \to y} \frac{x^k-y^k}{x-y} = k*y^{k-1}$ for $y\in\mathbb R$ and $k\in\mathbb N$

  42. Monotone homeomorphism on $[0,1]$
  43. interesting question about subsequential limits

  44. Let $f:[0,1]\to [0,\infty)$ be a continuous function. If $\int_0^1 f\, dx =0$ then $f(x)=0$ for all $x$.

  45. The given set is closed in $\mathbb{R}$?

  46. Find probability density function for random variable of fair coin toss
  47. Affine change of variables to prove that the Poission integral is harmonic?

  48. How can we check the continuity of that function?
  49. a Cauchy sequence converges
  50. Proof completeness of $(X,d)$ given $d$ is French railway metric

  51. General topology, prove or disprove.

  52. Interchanging the limit with the Riemann-Stieltjes Integral
  53. Is this proof on closed sets correct?
  54. Convex function as supremum

  55. Let $(a_n)$ be a sequence of positive real numbers such that $\sum\limits_{n=1}^{\infty} a_n$ is convergent.
  56. Establishing monotonicity of the sequence of ratio of Fibonacci like sequence

  57. countable additivity for a collection of algebra imply countable subadditivity

  58. Exercises in Sterling K. Berberian's Linear Algebra

  59. Trouble understanding pointwise vs uniform boundedness

  60. If $f'(x_0)>0$, how do I explicitly show there is a neighborhood of $x_0$ in which the difference quotient is strictly positive?

  61. Martingale Maximal function Inequality

  62. Does $\lim_{n\rightarrow\infty}f\left(nx\right)=0\quad\forall x>0$ imply $\lim_{x\rightarrow+\infty}f\left(x\right)=0$?
  63. Proving the Marcinkiewicz Interpolation Theorem

  64. Showing the existence of subsequences

  65. On Urysohn's lemma
  66. Prove that $\sum_{i=0}^d {n\choose i}\leq n^d +1$

  67. Nonatomicity and continuity

  68. Lifting functions from co-spherical to spherical bundle

  69. Let $A:l_1 \to c_0 : (x_k)_{k=1}^\infty \to (x_k^2)_{k=1}^\infty$. Is the operator A injective, surjective, continous, homeomorphism?

  70. Continuous $f:[0,1]\to\mathbb{R}$ such that $f(0)=f(1)$ and $\forall\alpha\in(0,1)\exists c\in[0,1-\alpha]|f(c)=f(c+\alpha)$?

  71. Prove there is a $\delta >0$ such that $\rho(x,y) \geq \delta$ for all $x \in K$,$y \in F$. Prove this may fail if $K$ is closed, but not compact.
  72. Solving the Differential equation: $y'=\frac{2}{x}y+x^3$

  73. $(Ax)(t)= \int_{0}^{1} \frac{\sin(ts)}{\mid t-s \mid^{\frac{1}{3}}} x(s) ds$ for $A : L_2[0,1] \to L_2[0,1] $ is compact.
  74. Scaling a Lebesgue measurable set average-wise

  75. Conditions for Inverse Function Theorem
  76. Fourier series of a 2-D function that satisfies $c(t,s) = c(|t-s|)$
  77. How to show that $f'(x)<2f(x)$
  78. limit of $\frac{(2n)!}{4^n(n!)^2}$

  79. Existence of the Dirac δ-function defined as a distribution?

  80. Proving convergent sequences are Cauchy sequences

  81. Fourier series of $\arccos(\lambda\cos x$)
  82. Is $h(x)=g(x) \circ f^{-1}(x)$ twice differentiable when $g$ and $f$ is?
  83. Example for a continuous function that has directional derivative at every point but not differentiable at the origin

  84. $f$ is of $C^2$ therefore $g\circ f$ is of $C^2$, for all functions $g$?

  85. If $f: \mathbb{R}\rightarrow\mathbb{R_+}$ is concave and strictly increasing, must there be an $A \in \mathbb{R}$ so that $df/dx|_{x=A}<\epsilon$

  86. If $\lim_{x\to \infty} f(x)<0$, must there exist some $x'\in \mathbb{R}$ so that $f(x')<0$?

  87. Differential system $y'=M(x)y$
  88. Define $f : \mathbb{R}\to \mathbb{R}$ by setting $f(0) = 0$, and $f(t)=t^2\sin(1/t) \text{ if } t\neq 0$
  89. $f'(x)>c>0$ for all $x\in [0,\infty)$. Show that $\lim_{x\to \infty}f(x)=\infty.$

  90. Is $f(0)$ finite?

  91. Optimization problem subject to an equality constraint

  92. Show that $x^3+ax^2+b$ where $a>0$ does not have more than one root using Rolle's Theorem.

  93. finding minimum and maximum with equality constraints
  94. An approximation of the real part of $\int_0^{\pi/2}x\left(-1+\sin x\right)^{\log x} dx$
  95. Show that $\sum_{k=2}^\infty d_k$ converges to $\lim_{n\to\infty} s_{nn}$.

  96. If ${x_n}$ strictly decreasing and $\lim_{n\to+\infty}x_n=0$ then does $\sum_{n=1}^{\infty}\frac{x_n-x_{n+1}}{x_n}$ diverges?
  97. Graph of continuous function from $[0,1]$ to $[0,1]$.
  98. Lagrange multipliers with equality constraints
  99. Prove convergence / divergence of $\sum_{n=2}^\infty(-1)^n\frac {\sqrt n}{(-1)^n+\sqrt n}\sin\left(\frac {1}{\sqrt n}\right)$
  100. If f is uniformly continuous in R. Then g (x, y) = f (x) -f (y) is uniformly continuous in R^2