Question

Looking for a quick explanation why this function is continuous at a, but not differential. Picture attached.

Answer 1

a differential is defined as the limit from both sides matching

Answer 2

To be honest i'm not quite sure what differential means, would like a better explanation. I'm pretty sure i understand continuous.

Answer 3

what is the limit as x approaches to from the left?

what is the limit as x approaches to from the right?

are they the same?

Answer 4

Nope, one is -1 the other is 1. However that is my next question, i don't see where those answers came from.

Answer 5

taking a derivative means that we are finding the slope of a line that is tangent at a particular point; of the slope from the left and right match, we can then define the slope at the point

Answer 6

notice that the function is an absolute value function, so at some point the slope of the line changes drastically

Answer 7

Oh i see, so since they differ, it can't be differentiable?

Answer 8

correct, how would you define the slope of the line at x=a?

Answer 9

You can't? Because it depends on what side, or would it be 0?

Answer 10

This problem was pretty obvious, because i could find the slope just by looking. If it were more complicated, how would i use the derivative function to find the slopes for separate right and left sides though?

Answer 11

0 is definable, its a flat line

but spose you took a ruler and tried to match the slope of the line at that point. you would have no way of saying what it would be

Answer 12

if someone is coming from the left they say ,,, ahh, its going to be 1

someone measuring it from the right will argue and say .. nah, its -1

who is right?

Answer 13

Oh gotcha, wow that was a perfect way to explain it. Thanks

Answer 14

might have those backwards, but same idea :)

Answer 15

This problem was pretty obvious, because i could find the slope just by looking. If it were more complicated, how would i use the derivative function to find the slopes for separate right and left sides though?

Answer 16

Still wondering about this though. The rest makes sense!!

Answer 17

"well behaved" functions will be nice and curvy; no sudden cusps or corners or breaks in them.

also, certain rational functions (glorified fractions) have tricky spots to them:

Answer 18

anytime you see a piecewise function ... be wary

Answer 19

Alright, guess thats all i need to know?

Answer 20

fer now ... i suspect theyll change it up on me in about 32 years or so :)

Answer 21

haha of course they will

Answer 22

Thanks again : )

Answer 23

good luck :)