Show that there exist continuous functions $g(x)\leq f(x) \leq h(x)$ for all $x \in [0,1]$, and $\int_{0}^{1} (h(x)-g(x))dx < \epsilon$.
86618
Approximating Riemann integrable functions
33114
Uniqueness of a continuous extension of a function into a Hausdorff space
30036
Function in $H^1$, but not continuous
87239
Prove functions defined by sup and inf are continuous
21301
If nonnegative $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$, then $\int_0^1 \Big| \frac{f''(x)}{f(x)} \Big| \,dx >4$
126461
Show that $g:I\to\mathbb{R}$ is continuous and $f:(x-x_0)g(x)$ is differentiable
142494
Functions $f(x)/g(x), g(x)/h(x),h(x)/f(x)$ are constant
96273
Is a continuous function $ f\colon(0,\infty)\to R$, such that $f(x)\leq f(nx)$ increasing?
131992
A problem on continuous functions
75479
Proving that this function is continuous on $G\times G$
5419
Continuous functions with $f^2(x)=g^2(x)\neq 0$
59519
What about the continuity of these functions in the uniform topology?
50103
Show that a continuous function satisfying $f(2^x)=f(3^x)$ for $x\in (0,\infty)$ is a constant function
41201
How prove there is no continuous functions $f:[0,1]\to \mathbb R$, such that $f(x)+f(x^2)=x$.
97104
Proof that continuous function on $ [0, 2]$ has $f(x) = f(x-1)$ for some $ x$
17223
Continuity of an inverse function.
149399
Given continuous $f$ and $g$, there exists $c \in (a,b)$ such that $\int_{a}^{c} f(x) \ \mathrm dx = \int_{c}^{b} g(x) \ \mathrm dx$