<p>In this paper we investigate a conjecture of Janusz Matkowski concerning the continuous solutions of the functional equation <Equation ID="Equ8"> <EquationSource Format="TEX">\(\begin{aligned}\begin{aligned} f\big (f(-x)+x\big )=f\big (-f(x)\big )+f(x),\qquad x\in \mathbb {R}. \end{aligned}\end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>f</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mo>-</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>x</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mo>=</mo> <mi>f</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mo>-</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="2em" /> <mi>x</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>Matkowski conjectured that all continuous solutions must necessarily be linear on both the negative and the positive half-line. We show, however, that the family of continuous solutions to the equation in question is far richer than anticipated: there exist continuous solutions that admit an arbitrary part. In addition, we provide a sufficient condition which, in the continuous setting, enforces the conclusion predicted by Matkowski’s Conjecture.</p>