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March 26th, 2018, 10:13 AM   #11
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Which one do we take as RHD & LHD? Because to my question, answers are like is differentiable/ has L.H.D but not R.H.D/ has R.H.D but not L.H.D/ Neither L.H.D nor R.H.D.
If f(x) is not continuous when x = 0, is f(x) differentiable when x = 0?
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March 27th, 2018, 03:53 AM   #12
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If f(x) is not continuous when x = 0, is f(x) differentiable when x = 0?
Every differentiable function is continuous. So, it won't.
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March 27th, 2018, 04:39 AM   #13
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Originally Posted by Lalitha183 View Post
Which one do we take as RHD & LHD? Because to my question, answers are like is differentiable/ has L.H.D but not R.H.D/ has R.H.D but not L.H.D/ Neither L.H.D nor R.H.D.
"RHD" means "Right hand derivative"- that is, coming toward x from that right (above) so is $\displaystyle \frac{f(x+h)- f(x)}{h}$ with h positive, x+ h> x. "LHD" means "Left hand derivative"- that is, coming toward x from the left (below) so is $\displaystyle \frac{f(x+ h)- f(x)}{h}$ with h negative, x+ h< x.
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March 27th, 2018, 06:13 AM   #14
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Every differentiable function is continuous. So, it won't.
There you go.

Just remember that although

$\text {Not continuous } \implies \text { not differentiable}$,

$\text {Continuous } \not \implies \text { differentiable.}$

So you must address the existence and continuity of the Newton quotient after you find the function to be continuous. And like all limits, to be continuous, the limit of the Newton quotient must exist and be equal whether approached from left or right.
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