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Question Number 214342 by issac last updated on 06/Dec/24

why differantiable f → f is continious but f is continous ↛ differantiable ??

$$\mathrm{why} \\ $$$$\mathrm{differantiable}\:{f}\:\rightarrow\:{f}\:\mathrm{is}\:\mathrm{continious}\: \\ $$$$\mathrm{but}\:{f}\:\mathrm{is}\:\mathrm{continous}\:\nrightarrow\:\mathrm{differantiable}\:?? \\ $$

Commented by mr W last updated on 06/Dec/24

a smooth line is always continous, but a continous line is not always smooth.

$${a}\:{smooth}\:{line}\:{is}\:{always}\:{continous}, \\ $$$${but}\:{a}\:{continous}\:{line}\:{is}\:{not}\:{always} \\ $$$${smooth}. \\ $$

Commented by mr W last updated on 06/Dec/24

Commented by mr W last updated on 06/Dec/24

all are continious, but not all are smooth (differentiable).

$${all}\:{are}\:{continious},\:{but}\:{not}\:{all}\:{are} \\ $$$${smooth}\:\left({differentiable}\right). \\ $$

Answered by Frix last updated on 06/Dec/24

Just think of an example: f(x)=∣x∣ is continuous but not differentiable at x=0 g(x)=((x^2 −1))^(1/3) is continuous but not differentiable at x=±1

$$\mathrm{Just}\:\mathrm{think}\:\mathrm{of}\:\mathrm{an}\:\mathrm{example}: \\ $$$${f}\left({x}\right)=\mid{x}\mid\:\mathrm{is}\:\mathrm{continuous}\:\mathrm{but}\:\mathrm{not}\:\mathrm{differentiable} \\ $$$$\mathrm{at}\:{x}=\mathrm{0} \\ $$$${g}\left({x}\right)=\sqrt[{\mathrm{3}}]{{x}^{\mathrm{2}} −\mathrm{1}}\:\mathrm{is}\:\mathrm{continuous}\:\mathrm{but}\:\mathrm{not}\:\mathrm{differentiable} \\ $$$$\mathrm{at}\:{x}=\pm\mathrm{1} \\ $$

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