Real analysis: Show a continuous and nonnegative function on $[a, b]$ = 0
92089
If $f:[a,b]\to \mathbb{R}$ is continuous and nonnegative and $\int_a^b{f}=0$, then $f(x)=0$ for all $x\in [a,b]$
58664
Proving a function is continuous for a fixed variable
68048
Prove that if $f$ is differentiable on $[a,b]$ and $f'$ is monotonic on $[a,b]$ , $f'$ is continuous on $[a,b]$
21301
If nonnegative $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$, then $\int_0^1 \Big| \frac{f''(x)}{f(x)} \Big| \,dx >4$
18222
Set of continuity points of a real function
20091
Every continuous function is the difference of two nonnegative continuous functions
113503
Show that exists a not decreasing function that $f:(a,b)\rightarrow \mathbb{R}$ that is continuous only in $(a,b)\setminus D$.
83676
Condition for continuity of bilinear form
71487
Real Analysis: Continuity of a Composition Function
75735
Let $f$ be a continuous function on an interval around $0$ and let $a_i=f(\frac{1}{i})$ (for large enough $i$)
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 $\mathbb R$
54940
If integral is zero and function is continuous and non negative, then what about the function?
13224
Continuous function positive at a point is positive in a neighborhood of that point
105084
Let f be a function twice differentiable and with derivatives continuous on an interval $[a,b]$ containing $0$. Prove the following statement:
149095
$f(\mathbb{R}\setminus \mathbb{Q}) \subseteq \mathbb{Q}$ and $f(\mathbb{Q}) \subseteq \mathbb{R}\setminus \mathbb{Q}$ imply that $f$ is not continuous
5419
Continuous functions with $f^2(x)=g^2(x)\neq 0$
65033
Prove inverse of strictly monotone increasing function is continuous over the range of original function