Prove that $f(x):=\begin{cases}e^{-\frac{1}x},\,\text{if}\, x>0\\0,\,\text{otherwise}\end{cases}$ is of class $C^\infty$
7861
Non-analytic smooth function
30385
Proving that $\frac{e^x + e^{-x}}2 \le e^{x^2/2}$
111756
How do I prove that then $f(x) = 1$ for some $x ∈ [0, 1]$?
66406
How to calculate $f(x)$ in $f(f(x)) = e^x$?
13287
Let $\int_{- \infty}^{\infty} f(x) dx =1$. Then show that $ \int_{- \infty}^{\infty} \frac{1}{1+ f(x)} dx = \infty.$
84785
Is $f(x)=x+\frac{x}{x+1}$ uniformly continuous on $(0,\infty)$
156727
A step in the proof of Characterization of $W^{1,\infty}$
139111
Prove that $a<c<b$, $\frac{f(b)-f(a)-(b-a)f'(a)}{g(b)-g(a)-(b-a)g'(a)}=\frac{f"(c)}{g"(c)}$
158888
Prove that the function $f(x)=\frac{x}{x^2+1}$ is continuous at $x=-1$.
130322
Show that $f(x) = 0$ if $x = 0$ $f(x) = \text{exp}(-1/x^{2})$ if $x \neq 0$ is infinitely differentiable but not analytic at $x_{0} = 0$ `
2414
If $f \in \operatorname{Hol}(D)$, $f(\frac{1}{2}) + f(-\frac{1}{2}) = 0$, prove that $|f(0)| \leq \frac{1}{4}$
149586
If $\lim_{x\to\infty} f'(x) = L$, prove that $\lim_{x\to\infty} f(x+c)-f(x) = cL$ and $\lim_{x\to\infty} f(x)/x=L$
35325
Prove that $\lim_{n\to\infty}\frac1{n}\int_0^{n}xf(x)dx=0$.
67020
Proving that for a smooth function if $f(\frac 1 k)=0 :\forall k\in \mathbb N$ then $f(x)=0 :\forall x\in[-1,1]$
72693
Prove: $\lim_{x\to\infty}x^{2/3}((x+1)^{1/3}-x^{1/3})=1/3$
134787
How to prove $\lim \limits_{n\to\infty}(n+1)\int_{0}^{1}x^nf(x)dx=f(1)$
17646
If $\frac{f(x)}{x}$ is decreasing in $(0,\infty)$, prove that $f(a+b) \leq f(a) + f(b)$ for $a,b >0$