Show that $\frac{1}{x^x}$ is continuous on $[0,1]$, and find its integral on this range.
74454
Sophomore's dream: $\int_0^{1} x^{-x} \; dx = \sum_{n=1}^\infty n^{-n}$
19916
$f(x)=f(x^2+ 1/4)$ , $f$ is continuous from $\mathbb{R}$ to $\mathbb{R}$
69978
Evaluating $\int_1^\infty \frac{1}{x(x^2+1)}\ dx$
53349
Prove that $\sqrt{x}$ is continuous on its domain $[0, \infty).$
14578
Proving that an additive function $f$ is continuous if it is continuous at a single point
54797
Find the continuous function defined over $[0,1]$ such that $\int_0^1f(x){\rm d}x=1, \int_0^1 xf(x){\rm d}x=a$ and $\int_0^1x^2f(x){\rm d}x=a^2 .$
36166
Uniform continuity on (0,1) implies boundedness
78319
Show that $x \mapsto \frac{f(x)}{x}$ is strictly increasing on (0,1) given that f '(x) is strictly increasing on (0,1) and that f(0)=0
38885
Prove the function is continuous, exercise from Conway's "Functions of One Complex Variable I"
75735
Let $f$ be a continuous function on an interval around $0$ and let $a_i=f(\frac{1}{i})$ (for large enough $i$)
16775
Showing that $\int_{0}^{\infty} \frac{dx}{1 + x^2} = 2 \int_0^1 \frac{dx}{1 + x^2}$
135449
Prove: $\frac1x$ is not a uniformly continuous function
88640
Is $\sqrt x$ continuous at $0$? Because it is not defined to the left of $0$
158888
Prove that the function $f(x)=\frac{x}{x^2+1}$ is continuous at $x=-1$.
61782
How to prove that $f(x) = \frac{-2x+1}{(2x-1)^2-1}$ is one-to-one on $(0,1)$?
77943
If $f$ is continuous, then there exists $x+\frac{1}{n}\in[0,1]$ and $f\left(x+\frac{1}{n}\right)=f(x)$
60700
Continuity of a function to the integers