Answer to Question #347211 in Real Analysis for Nikhil

Question #347211

Every continuous function is differentiable.


True or false with full explanation.

1
Expert's answer
2022-06-03T12:51:47-0400

False.

Counterexample

The function f(x)=xf(x)=|x| is continuous on (,).(-\infin, \infin).


limΔx0f(0+Δx)f(0)Δx=limΔx0Δ0Δx=1\lim\limits_{\Delta x\to0^{-}}\dfrac{f(0+\Delta x)-f(0)}{\Delta x}=\lim\limits_{\Delta x\to0^{-}}\dfrac{-\Delta-0}{\Delta x}=-1

limΔx0+f(0+Δx)f(0)Δx=limΔx0+Δ0Δx=1\lim\limits_{\Delta x\to0^{+}}\dfrac{f(0+\Delta x)-f(0)}{\Delta x}=\lim\limits_{\Delta x\to0^{+}}\dfrac{\Delta-0}{\Delta x}=1

limΔx0f(0+Δx)f(0)Δx=1\lim\limits_{\Delta x\to0^{-}}\dfrac{f(0+\Delta x)-f(0)}{\Delta x}=-1

1=limΔx0+f(0+Δx)f(0)Δx\not=1=\lim\limits_{\Delta x\to0^{+}}\dfrac{f(0+\Delta x)-f(0)}{\Delta x}

Therefore


limΔx0f(0+Δx)f(0)Δx\lim\limits_{\Delta x\to0}\dfrac{f(0+\Delta x)-f(0)}{\Delta x}

does not exist.

Therefore the function f(x)=xf(x)=|x| is not differentiable at x=0.x=0.


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