On continuity results for some differential equations with Caputo-Fabrizio derivative

In this paper, we study a differential equation with the Caputo-Fabrizio derivative. First, for a given function $f$, we focus on the convergence results of the Caputo-Fabrizio derivative of $f$ and the first derivative of $f$. We also study the continuity results for fractional-order Caputo-Fabrizio derivative. The second result is considering continuity results for the mild solution to linear problems. The third result mentions the continuity results for the mild solution to nonlinear problem. Additionally, numerical examples are investigated to verify some of the theoretical result.