I am trying to find a function $f: \mathbb{R}^+ \to \mathbb{R}^+$ that fullfils the following conditions

• $$f \in \mathcal{C}^1(\mathbb{R}^+,\mathbb{R}^+)$$

• $$\int_{\mathbb{R}^+} f \in \mathbb{R}^+$$

• $$\int_{\mathbb{R}^+} \mid f' \mid \in \mathbb{R}$$

• $$f \text{is not $\frac{1}{2}$-Hölder}$$

I've tried functions with smooth spikes but I am unable to express this function as combinations of usual functions.

Moreover, It's worth noticing that if we have the assumption : $f'^2$ is integrable then $f$ is necessarily $\frac{1}{2}-$Hölder :

We have (using CS) : $$ \mid f(x) -f(y) \mid \leq \int_x^y 1 \times f' \leq \sqrt{\int_x^y f'^2}\sqrt{y-x} $$ Hence it follows that $f$ is $\frac{1}{2}-$Hölder continous since that $\sqrt{\int_x^y f'^2}$ is bounded.