Study the variations of the function $t \mapsto t \ln(t)$ on $\mathbf{R}_{+}^{*}$. Verify that the function can be extended by continuity at 0 and verify that the quantity $\operatorname{Ent}_{\varphi}(g)$ is well defined for all $g \in C^{0}(\mathbf{R}) \cap CL(\mathbf{R})$ with strictly positive values such that $\int_{-\infty}^{+\infty} g(x) \varphi(x) \mathrm{d}x = 1$. Recall that for $f \in C^{0}(\mathbf{R}) \cap CL(\mathbf{R})$ with strictly positive values such that $\int_{-\infty}^{+\infty} f(x) \varphi(x) \mathrm{d}x = 1$, the entropy of $f$ with respect to $\varphi$ is defined by: $$\operatorname{Ent}_{\varphi}(f) = \int_{-\infty}^{+\infty} \ln(f(x)) f(x) \varphi(x) \mathrm{d}x$$
Study the variations of the function $t \mapsto t \ln(t)$ on $\mathbf{R}_{+}^{*}$. Verify that the function can be extended by continuity at 0 and verify that the quantity $\operatorname{Ent}_{\varphi}(g)$ is well defined for all $g \in C^{0}(\mathbf{R}) \cap CL(\mathbf{R})$ with strictly positive values such that $\int_{-\infty}^{+\infty} g(x) \varphi(x) \mathrm{d}x = 1$.
Recall that for $f \in C^{0}(\mathbf{R}) \cap CL(\mathbf{R})$ with strictly positive values such that $\int_{-\infty}^{+\infty} f(x) \varphi(x) \mathrm{d}x = 1$, the entropy of $f$ with respect to $\varphi$ is defined by:
$$\operatorname{Ent}_{\varphi}(f) = \int_{-\infty}^{+\infty} \ln(f(x)) f(x) \varphi(x) \mathrm{d}x$$