How to prove $f(x) = \ln x$ continuous by proving first that $f(x)$ continuous at $1$, and then by using $\ln (xy) = \ln(x) + \ln(y)$.
71049
Functional equation $f(xy)=f(x)+f(y)$ and continuity
144589
Proving that if $f(x) \le g(x)$ then $f(x_0) \le g(x_0)$
3518
Proving $f(x)=1/x$ on $(0,1 )$ is not uniformly continuous
11879
If $\lim_{n\rightarrow\infty} f(n+1) - f(n) = L$, prove that $\lim_{n\rightarrow\infty} f(n)/n = L$
37966
If $f$ is continuous and $f(x+y) = f(x)+f(y)$, then $f(x) = cx$ for all $x \in \mathbb{R}$
104688
How can I prove that $x-{x^2\over2}<\ln(1+x)$
48275
$f$ defined $|f(x) - f(y)| \leq |x - y|^{1+ \alpha}$ Prove that $f$ is a constant.
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
94781
Prove that $M(t)=\sup_ {a \leq x \leq t} f(x)$ given $f(x)$ is continuous on $[a,b]$
97256
For $f$ continuous, show $\lim_{n\to\infty} n\int_0^1 f(x)x^n\,dx = f(1).$
154034
Suppose that $f(x)$ is continuous on $(0, \infty)$ such that for all $x > 0$,$f(x^2) = f(x)$. Prove that $f$ is a constant function.
143174
property of continuous function
113687
$f$ is continuous function from $ [0,1]$ to itself and f(0)=0, f(1)=1 and $f(f(x))=x$ then show that $f(x)=x$ on$[0,1]$
130023
Prove that the sum of two continuous functions is continuous
83494
Uniform continuity of $\ln(x)$
95970
Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function. If $f(0)=f(1)$ then exists $x\in [0,1]$ s.t. $f(x) = f(x+\frac{1}{2})$
153730
Show that $f(x)=x^2$ is continuous at $a=2$ using the $\delta-\epsilon$ definition of continuity.