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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:Â 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) }](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Clim+_%7B+x%5Clongrightarrow+c+%7D%7B+%5Cleft%5B+%5Cleft%28+%5Cfrac+%7B+f%5Cleft%28+x+%5Cright%29+-f%5Cleft%28+c+%5Cright%29+%7D%7B+x-c+%7D+%5Cright%29+%5Cright%5D+%7D+%5C%5C+Let%5Cquad+%5Clim+_%7B+x%5Clongrightarrow+c+%7D%7B+%5Cleft%28+%5Cfrac+%7B+f%5Cleft%28+x+%5Cright%29+-f%5Cleft%28+c+%5Cright%29+%7D%7B+x-c+%7D+%5Cright%29+%3Df%27%28c%29%5Cqquad+.......%28i%29+%7D++&bg=ffffff&fg=000&s=1&c=20201002)
In order to prove that f(x) is continuous at x=c, it is sufficient to show that
Hence f(x) is continuous at x=c
![\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)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%7B+f%5Cleft%28+x+%5Cright%29+%7D+%3Df%5Cleft%28+c+%5Cright%29%5C%5C%5Clim+_%7B+x%5Clongrightarrow+c+%7D%7B+f%5Cleft%28+x+%5Cright%29+%7D+%3D%5Clim+_%7B+xc+%7D%7B+%5Cleft%5B+%5Cleft%28+%5Cfrac+%7B+f%5Cleft%28+x+%5Cright%29+-f%5Cleft%28+c+%5Cright%29+%7D%7B+x-c+%7D+%5Cright%29+%5Cleft%28+x-c+%5Cright%29+%2Bf%5Cleft%28+c+%5Cright%29+%5Cright%5D+%7D+%5C%5C+%3D%5Clim+_%7B+x%5Clongrightarrow+c+%7D%7B+%5Cleft%5B+%5Cleft%28+%5Cfrac+%7B+f%5Cleft%28+x+%5Cright%29+-f%5Cleft%28+c+%5Cright%29+%7D%7B+x-c+%7D+%5Cright%29+.%5Cleft%28+x-c+%5Cright%29+%5Cright%5D+%7D+%2Bf%5Cleft%28+c+%5Cright%29+%5C%5C+%3D%5Clim+_%7B+x%5Clongrightarrow+c+%7D%7B+%5Cfrac+%7B+f%5Cleft%28+x+%5Cright%29+-f%5Cleft%28+c+%5Cright%29+%7D%7B+x-c+%7D+%7D+.%5Clim+_%7B+x%5Clongrightarrow+c+%7D%7B+%5Cleft%28+x-c+%5Cright%29+%2B+%7D+f%5Cleft%28+c+%5Cright%29+%5C%5C+%3D%5Clim+_%7B+x%5Clongrightarrow+c+%7D%7B+f%5Cleft%28+x+%5Cright%29+%7D+%3Df%5Cleft%28+c+%5Cright%29++&bg=ffffff&fg=000&s=1&c=20201002)
Converse:Â 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
