What is the difference between differentiability and continuity of a function?

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awalkbyfaithbzu6 · Accepted Answer

A function&amp;nbsp;f(x)&amp;nbsp;is continuous in the point&amp;nbsp;x0&amp;nbsp;if the limit:limx→x0f(x)exists, is finite, and has value equal to the function:f(x0)=limx→x0f(x)A function f(x) is differentiable in the point&amp;nbsp;x0&amp;nbsp;if the limit:f′(x0)=limx→x0f(x)−f(x0)x−x0exists and is finite.Any continuous function is a differentiable function.We can prove it by writing&amp;nbsp;f(x)&amp;nbsp;as:f(x)=f(x0)+f(x)−f(x0)x−x0(x−x0)Passing to the limit for&amp;nbsp;x→&amp;nbsp;x0:limx→x0f(x)=f(x0)+limx→x0(f(x)−f(x0)x−x0)⋅limx→x0(x−x0)limx→x0f(x)=f(x0)+f′(x0)⋅0=f(x0)Continous but not differentiable functions include, for instance:y=|x|is continuous but not differentiable in&amp;nbsp;x=0.

libertydragonrbha · Accepted Answer

If a function is differentiable, then every point on its graph exhibits a slope. ... The absolute value of x function is a continuous function because it doesn&#039;t break up, which is a requirement for a function to be continuous.

What is the difference between differentiability and continuity of a

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