Find functions f(x) continuous on the real line satisfying given conditions

Find functions $f(x)$ continuous everywhere on $\mathbb{R}$ such that

$f(x) = 2 + \int_1^x f(t) \, dt.$

Let

$y = \int_1^x f(t) \, dt.$

Then $y' = f(x)$ so we are trying to find solutions of the differential equation

$y' = 2 + y.$

We could use theorems from previous sections to solve this, but it is also a separable first-order equation so we can solve it using the techniques of this section,

$\begin{align*} y' = 2 + y && \implies && \frac{y'}{2+y} &= 1 \\ && \implies && \int \frac{1}{2+y} \, dy &= \int \, dx \\ && \implies && \log |2+y| &= x + C \\ && \implies && 2 + y &= Ce^x. \end{align*}$