<p>In this paper, firstly, a new definition of a fuzzy difference and a fuzzy addition on non-increasing and left continuous fuzzy real number is given, which is different from the previous approach by using Zadeh’s extension principle and fuzzy interval-numbers. Also, we give a reasonable construction of the distance between non-increasing and left continuous fuzzy real numbers. Based on these, we introduce the definitions and properties of the fuzzy limit and the fuzzy continuity on fuzzy real-valued functions. What is more, the concept of a fuzzy multiplication is presented, for the sake of introducing a novel definition of a fuzzy derivative on fuzzy real-valued functions, which is the main purpose of this paper. To be specific, <i>f</i> is differentiable at <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2024_3064_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(x_0\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>x</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> if and only if the limits <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2024_3064_Article_IEq2.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="229" /> </InlineMediaObject> <EquationSource Format="TEX">\( \lim _{h\rightarrow 0^{+}}\frac{1}{h}\odot (f(x_0+h)\ominus f(x_0))\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo movablelimits="true">lim</mo> <mrow> <mi>h</mi> <mo stretchy="false">→</mo> <msup> <mn>0</mn> <mo>+</mo> </msup> </mrow> </msub> <mfrac> <mn>1</mn> <mi>h</mi> </mfrac> <mo>⊙</mo> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> <mo>⊖</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2024_3064_Article_IEq3.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="230" /> </InlineMediaObject> <EquationSource Format="TEX">\( \lim _{h\rightarrow 0^{+}}\frac{1}{h}\odot (f(x_0)\ominus f(x_0-h))\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo movablelimits="true">lim</mo> <mrow> <mi>h</mi> <mo stretchy="false">→</mo> <msup> <mn>0</mn> <mo>+</mo> </msup> </mrow> </msub> <mfrac> <mn>1</mn> <mi>h</mi> </mfrac> <mo>⊙</mo> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>⊖</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>-</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> both exist and equal, where the symbols <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2024_3064_Article_IEq4.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(\odot \)</EquationSource> <EquationSource Format="MATHML"><math> <mo>⊙</mo> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2024_3064_Article_IEq5.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(\ominus \)</EquationSource> <EquationSource Format="MATHML"><math> <mo>⊖</mo> </math></EquationSource> </InlineEquation> represent the novel definition of fuzzy multiplication and fuzzy difference.</p>