The paragraph says, " Even if the function f is an everywhere differentiable function, it is still possible for f ' to be discountinuous. However, the graph of f ' can never exhibit a discountinuity of ...." picture is in paint document...(adsbygoogle = window.adsbygoogle || []).push({});

What type of discountinuity is that? a hole discountinuity?

second Question: Is my understanding correct? This is my explanation why f is not differentiable at some x value.

Given

f ' (x) =

x , x>0

x^2 x<0

7 , x = 0.

Therefore, the graph of the differnetiable function should have a similar discontinuity as the one shown in the paint document. (Determining the limit)

lim f ' (x) as x → 0- = lim f ' (x) as x → 0+ = 0 however because lim Δx → 0 [ f(0 + Δx) - f(0) ]/Δx is equal to f ' (0) = 7 ≠ 0, therefore the function f non differentiable at x = 0 because inorder for f to be diff at 0, f'(0) must equal 0

Is my understanding why f is not differentiable at x = 0 correct?

Next, If I wanted to make f differentiable at x could I do this by defining one of two functions....

f ' (x) =

x , x>0

x^2 x<0

0 , x = 0.

OR

f ' (x) =

x , x>0

x^2 x<0

Would this work to make f diff at x = 0?

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# Homework Help: Limit Questioin

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