Let $f:\mathbb R\to \mathbb R$ be continuous, $f(x)\to+\infty$ as $x\to\pm\infty$. Show that $f$ has a minimum.
111290
If $f$ continuous and $\lim_{x\to-\infty }f(x)=\lim_{x\to\infty }f(x)=+\infty $ then $f$ takes its minimum.
126259
Minimum of a function on an open interval
122160
$f:\mathbb R \to \mathbb R$ be a continuous such that $f(f(f(x)))=x,\forall x \in \mathbb R$ , is $f(x)=x , \forall x \in \mathbb R$?
113503
Show that exists a not decreasing function that $f:(a,b)\rightarrow \mathbb{R}$ that is continuous only in $(a,b)\setminus D$.
91398
Continuous function on R which tends to infinity on open interval must have a minimum
47
Let $f$ be a continuous and open map from $\mathbb R $ to $\mathbb R$.Prove that $f$ is monotonic.
53627
Proving continuity of $f$
74950
Show a function for which $f(x + y) = f(x) + f(y) $ is continuous at zero if and only if it is continuous on $\mathbb R$
79107
Give an example of a function $f: \mathbb{R} \to \mathbb{R}$ which is continuous only at $0$.
46101
If $f(x)$ is continuous on $[a,b]$ and $M=\max \; |f(x)|$, is $M=\lim \limits_{n\to\infty} \left(\int_a^b|f(x)|^n\,\mathrm dx\right)^{1/n}$?
145984
if $f: (0,\infty) \to (0,\infty)$ is a strictly decreasing then $f \circ f$ is decreasing?
132824
Does $f$ monotone and $f\in L_{1}([a,\infty))$ imply $\lim_{t\to\infty} t f(t)=0$?
93021
$f:X\longrightarrow\mathbb{R}$ is continuous iff $\{x\in X:f(x)\geq\alpha\}$ and $\{x\in X:f(x)\leq\alpha\}$ are closed $\forall\alpha\in\mathbb{R}$
58597
A characterization of functions from $\mathbb R^n$ to $\mathbb R^m$ which are continuous
95429
If $f$ is continuous with $ \int_0^{\infty}f(t)\,dt<\infty$ then which are correct?
19072
$\int_{\mathbb R} |f(x)| dx < \infty \implies \int^{\infty}_{n} |f(x)| dx \to 0$ as $n \to \infty$?
108060
Suppose $f:[0,\infty]\rightarrow$ to R and has continuous $f'(x)$ and $f''(x)$ f(x) → 0 as x → ∞. Show that f′(x)→0 as x→∞