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Relation between Continuity and Differentiability

In previous section we have discussed that if a function is differentiable at a point, then it should be continous at that point and a discontinuous function cannot be differentiable. This fact is proved in following theorem

Theorem: If a function is differentiable at a point, it is necessarily continuous at that point.But the converse is not necessarily continuous true.

Or

f(x) is differentiable at x=c  f(x) is continuous at x=c

\displaystyle \lim _{ x\longrightarrow c }{ \left[ \left( \frac { f\left( x \right) -f\left( c \right) }{ x-c } \right) \right] } \\ Let\quad \lim _{ x\longrightarrow c }{ \left( \frac { f\left( x \right) -f\left( c \right) }{ x-c } \right) =f'(c)\qquad .......(i) }

In order to prove that f(x) is continuous at x=c, it is sufficient to show that

\displaystyle { f\left( x \right) } =f\left( c \right)\\\lim _{ x\longrightarrow c }{ f\left( x \right) } =\lim _{ xc }{ \left[ \left( \frac { f\left( x \right) -f\left( c \right) }{ x-c } \right) \left( x-c \right) +f\left( c \right) \right] } \\ =\lim _{ x\longrightarrow c }{ \left[ \left( \frac { f\left( x \right) -f\left( c \right) }{ x-c } \right) .\left( x-c \right) \right] } +f\left( c \right) \\ =\lim _{ x\longrightarrow c }{ \frac { f\left( x \right) -f\left( c \right) }{ x-c } } .\lim _{ x\longrightarrow c }{ \left( x-c \right) + } f\left( c \right) \\ =\lim _{ x\longrightarrow c }{ f\left( x \right) } =f\left( c \right)
Hence f(x) is continuous at x=c

Converse: The converse of above theorem is not necessarily true i.e., a function may be continuous may be continuous at a point but may not be differentiable at that point.

For example: The function f(x) =|x| is continuous at x=0 but it is not differentiable at x=0, as shown in the figure.

Which shows we have sharp edge at x=0 hence, not differentiable but continuous at x=0

April 18, 2019
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