Prove that $f'(x)=0$ if $|f'(x)|\leq M|f(x)-f(a)|$ where $M>0$ and $f:[a,b]\rightarrow\mathbb{R}$ is continuous
120600
Real analysis derivate problem. How can I prove this?
69008
$f'(x) = [f(x)]^{2}$. Prove $f(x) = 0 $
40915
Proof if $f$ continuous in $x_0$ then there is a neighbourhood of $x_0$ so f bounded.
13927
Values of $a$ s.t. for all continuous $f$ with $f(0)=f(1)$ there exists $x$ s.t. $f(x+a) = f(x)$
104263
How to show that $f'(x)<2f(x)$
150554
proof that $f'$ is continuous at $a$
67292
Prove that $\lim_{t\to \infty} t\mu(\{x:f(x)\geq t\})=0$
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.
105084
Let f be a function twice differentiable and with derivatives continuous on an interval $[a,b]$ containing $0$. Prove the following statement:
9957
Prove if $f:[a,b] \rightarrow \mathbb{R}$ is integrable, then so is $|f|$ and $\left|\int_{a}^{b}f(x)dx\right| \leq \int_{a}^{b}|f(x)|dx$
118207
Prove that the product of two continuous functions is continuous
69695
$f\geq 0$, continuous and $\int_a^b f=0$ implies $f=0$ everywhere on $[a,b]$
16212
If $f$ is continuous, nonnegative on $[a, b]$, show that $\int_{a}^{b} f(x) d(x) = 0$ iff $f(x) = 0$
22303
How do I prove that $f(x)f(y)=f(x+y)$ implies that $f(x)=e^{cx}$, assuming f is continuous and not zero?
104510
For continuous $f$, $\int_0^x f(t)dt = ∫_x^1f(t)dt$ implies $f=0$.
129551
Does $\lim_{h\rightarrow 0}\ [f(x+h)-f(x-h)]=0$ imply that $f$ is continuous?
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$[0,1]$