Suppose $f$ is continuous on R such that $\lim_{h\to 0} \frac{f(x+h)-f(x-h)}{h} = 0$ for all $x\in\mathbb R$. Prove that $f$ is constant.
128719
Are constants the only continuous functions with "symmetric derivative" zero?
86517
If $|f(z)|\le |g(z)|$, can we say $f$ is constant?
91352
If $\Re(f)$ is bounded then f is constant.
143174
property of continuous function
84510
If $f(\mathbb{C})\subset \mathbb{C}-[0,1]$ then $f$ is constant
90565
If $f:[0,1] \rightarrow \mathbb{R}$ is continuous function such that $f(0)=f(1)$ then there exists $ x\in [0,1]$ such that $f(x) = f(x+\frac{1}{n})$.
158152
Given $f: \Bbb R\rightarrow\Bbb R$ and a point $a\in\Bbb R$. Prove $lim_{x\rightarrow a} f(x)=\lim_{h\rightarrow 0} f(a+h)$ if 1 of the limits exists.
142637
If $f \in \mathcal{L}^1(\mathbb{R})$ then $\lim_{h\rightarrow 0}\int_{-\infty}^\infty |f(x+h)-f(x)|dx\rightarrow 0$
53627
Proving continuity of $f$
47381
If $\lim_{x\to\infty} [f(x+1)-f(x)] =l$ then $\lim_{ x\to\infty}f(x)/x =l$ ($f$ is continuous)
28399
Showing $f$ is constant
113871
Prove $f(x) = \sqrt x$ is uniformly continuous on $[0,\infty)$
137527
if $f(x)$ is differentiable at a x, prove that: $\lim_{h\to0}\frac{f(x+h)-f(x-h)}{2h}$
50103
Show that a continuous function satisfying $f(2^x)=f(3^x)$ for $x\in (0,\infty)$ is a constant function
69869
if $f$ is continuous on $\mathbb{R}$ and $f(r)=0,r \in \mathbb{Q}$, then $f(x)=0,x \in \mathbb{R}$
129579
If $f''(x_0)$ exists then $\lim_{x \to x_0} \frac{f(x_0+h)-2f(x_0)+f(x_0-h)}{h^2} = f''(x_0)$
105700
Proving $\,f$ is constant.