If there was a hole in the line at (2,3)

wuntsongo0cy

wuntsongo0cy

Answered question

2022-04-21

If there was a hole in the line at (2,3) and there is another point at (2,1), then would the graph be differentiable at that point and why?

Answer & Explanation

kubistiedt

kubistiedt

Beginner2022-04-22Added 17 answers

The short answer is that the function you have described is not continuous at 2. It is a theorem that if f is differentiable at c, then f is continuous at c. Therefore non-continuous implies non-differentiable.
Longer answer
"A hole in the line at (2,3)" indicates to me that limx2f(x)=3.
The point at (2,1) implies that f(2)=1
Now
f(2)=limx2f(x)f(2)x2
=limx2f(x)1x2
This limit has the form 3122=20 which entails that the limit does not exist.
Since the derivative is the limit, the derivative also does not exist.

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