find a positive continuous function with a finite area : $\int_0^\infty f(x) dx$ , but the $f(x)\rightarrow$ doesn't exist.
124997
finding examples for a non negative and continuous function for which the infinite integral is finite but the limit at infinity doesn't exist
128911
Continuous extension of a real function
17223
Continuity of an inverse function.
44269
If $f$ is nonnegative and continuous on $[a,b]$, then $\left(\int_a^b f(x)^n \ dx\right)^{1/n}\to\max\limits_{[a,b]} f$
58816
Existence of a continuous function which does not achieve a maximum.
122073
Computing $\lim_{n \rightarrow\infty} \int_{a}^{b}\left ( f(x)\left | \sin(nx) \right | \right )$ with $f$ continuous on $[a,b]$
126878
Monotonic function satisfying darboux property $\Rightarrow$ continuous
26329
Is there a shape with infinite area but finite perimeter?
33337
$\int_{0}^{\infty} f(x) \,dx$ exists. Then $\lim_{x\rightarrow \infty} f(x) $ must exist and is $0$. A rigorous proof?
13506
Can an unbounded function have a finite integral?
120947
continuous real-valued function
57014
Existence of a power series converging non-uniformly to a continuous function
148931
Constructing a function that is continuous and has a max on an open interval, but is not necessarily increasing immediately to the left of the max.
60700
Continuity of a function to the integers
776
Entire function with positive real part is constant (no Picard)
97256
For $f$ continuous, show $\lim_{n\to\infty} n\int_0^1 f(x)x^n\,dx = f(1).$
117486
Limit of uniformly continuous functions with convergent improper integrals.