My attempt...    By definition, whenever $|x- x_0| &lt; \delta$ we have $|f(x) - f(x_0)| &lt; \epsilon$. Observing that    \begin{align}  |f(x) - f(y)| &amp;= |f(x -x_0 + x_0) + f(y)| = |f(x-x_0) + f(x_0) - f(y)|  \newline \newline &amp;\leq \epsilon + |f(y) - f(x_0)| = \epsilon + |f(y-x_0)|... \end{align}      Here I need to choose a delta that can depends on $\epsilon$ and $y$ s.t. whenever $0&lt;|x-y|&lt; \delta$ then the above inequality is bounded by any $\epsilon$.     I'm also, in general, having trouble understanding this concept of continuity on an interval. I believe the structure of the definition is: for any $\epsilon&gt; 0$ and any number $y$ in the interval, there exists a $\delta$ that depends on $\epsilon$ and $y$ such that for all $x$ in the interval and $|x - y | &lt; \delta$ then $|f(x) - f(y)| &lt; \epsilon$.     This definition makes me tempted to just choose y to be in the same delta neighborhood as $x$ in the given statement, but that constricts continuity to a small interval.    Edit: This question assumes no knowledge of Lebesgue measure theory.
The  asks about functions $f \colon \mathbb R \to \mathbb R$ such that $$f(x+y)=f(x)+f(y).$$ It is a very well-known functional equation, which appears in various areas of mathematics ranging from exercises in freshman classes to constructing useful counterexamples for some advanced questions. Solutions of this equation are often called . Also a few other equations related to this equation are often studied. (Equations which can be easily transformed to Cauchy functional equation or can be solved by using similar methods.) Is there some overview of basic facts about Cauchy equation and related functional equations - preferably available online?
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