# Is a function differentiable at a point discontinuity?

I recently learnt the proof that if a function is differentiable then it is continuous and that a function cannot be differentiable at a point discontinuity . I am confused however take the function $$f(x)=\begin{cases} -2x & x<4, \\ 4 & x=9. \end{cases}$$ Isn't this function differentiable at $$x=9$$ as the left hand derivative exsists and the right hand derivative does not need to exsist as the function is not even defined for values of x greater than $$9$$ so what is the function differentiable and if not why?

• Did you mean $x<9$? Dec 30 '20 at 17:06
• Sure your function is right? As written, f is not defined between 4 and 9 so it makes no sense to talk about it being differentiable or continuous at x = 9.
– Paul
Dec 30 '20 at 17:09
• But isn't that the purpose of piecewise functions Dec 30 '20 at 17:11
• For calculating the left hand derivative, you would need the value of the function at the left of $9$. Remember that LHD is given by $$\lim_{h\to0}\frac{f(9)-\color{red}{f(9-h)}}h$$ Dec 30 '20 at 17:14
• As written, this function is continuous. For $x<4$, choose $\delta<\frac{\varepsilon}{2}$. For $x=9$, choose $\delta<5$, not depending on $\varepsilon$. But it's not differentiable at $9$, since differentiability only makes sense at limit points of the domain, and $9$ is not a limit point of $(-\infty,4)\cup\{9\}$. Dec 30 '20 at 17:35

This function is continuous but not differentiable at 9, this all comes down to the function's domain which is $$(-\infty, 4)\cup \{9\}$$. Here, 9 is an isolated point, and by the definition of continuity, every function is continuous at the isolated points of its domain. Moreover, as 9 is an isolated point, it is not an accumulation point, so again by definition you cannot calculate the limit $$\lim_{h\rightarrow 0}\dfrac{f(9+h)-f(9)}{h}$$. In fact, most books only define the derivative at a point when it is an interior point of the domain.
• You may wish to prepare yourself for a moving target. I suspect that the OP intended that $f(4) = 9,$ not $f(9) = 4.$ Dec 30 '20 at 17:52
• Oh, I see. In that case, f is not differentiable at 4, because $\lim_{h\rightarrow 0^-}\dfrac{f(4+h)-f(4)}{h}=\lim_{h\rightarrow 0^-}\dfrac{-8-2h-9}{h}=\lim_{h\rightarrow 0^-}\dfrac{-17}{h}-2=+\infty$. Dec 30 '20 at 18:06
• right, but arguably premature. examining the wording of the OP's query: [1] on the one hand, his inclusion of an isolated point in the domain is bizarre, and the inferred typo is plausible [2] "Isn't this function differentiable at $x=9$ as the left hand derivative exists" : he specifically referred to $x=9$ : is this another typo? [3] he said : "as the left hand derivative exists" : exists where, at $x=9$?? You may wish to hold off firing torpedoes until the target stops moving. Dec 30 '20 at 18:13