Prove that a continuous function has a minimum value if $\lim\limits_{x\to +\infty} f(x) = \lim\limits_{x\to -\infty} f(x) = +\infty$.
111290
If $f$ continuous and $\lim_{x\to-\infty }f(x)=\lim_{x\to\infty }f(x)=+\infty $ then $f$ takes its minimum.
117906
Show that $\liminf\limits_{x\to \infty}|f'(x)|=0$ if $\lim\limits_{x\to \infty}f(x)/x=0$ and $f$ is differentiable
9614
How does the existence of a limit imply that a function is uniformly continuous
20404
Prove that $f$ continuous and $\int_a^\infty |f(x)|\;dx$ finite imply $\lim\limits_{ x \to \infty } f(x)=0$
14896
Prove: the set of zeros of a continuous function is closed.
97316
Let $F : X → X$ be continuous. Prove that the set $\{x ∈ X : F(x) = x\}$ of fixed points of F is closed in X
97104
Proof that continuous function on $ [0, 2]$ has $f(x) = f(x-1)$ for some $ x$
143174
property of continuous function
57023
If $f,g$ are continuous at $a$, show that $h(x)=\max\{f(x),g(x)\}$ and $k(x)=\min\{f(x),g(x)\}$ are also continuous at $a$
145368
How to show $f:Y\to X^{+}$ is continuous?
25592
If a derivative of a continuous function has a limit, must it agree with that limit?
65827
When $\min_{x \in X,y \in Y} f(x,y) = \min_{x \in X} \min_{y \in Y} f(x,y)$?
126259
Minimum of a function on an open interval
125754
Proving or disproving a statement about continuity and limits
82280
Problem about continuous functions and the intermediate value theorem
3772
About the continuity of the function $f(x) = \sum\limits_k2^{-k}\mathbf 1_{q_k \leq x}$
118207
Prove that the product of two continuous functions is continuous