Question

How can you tell if a function is differentiable without graphing it?

Answer 1

there are some classes of functions that are known to be differentiable. For example all polynomials are differentiable.

Answer 2

yes, I remember that! c:
Any other rules?

Answer 3

So you can first see if it fits into a class of functions that are known to be differentiable.

Answer 4

\[\frac{ x+1 }{ x }\]

Answer 5

In particular, I was wondering how I could tell if this function is differentiable? And what about continuity?

Answer 6

Well that function has a discontinuity at x=0. So if you wondered how much the function is changing at that point, it would be undefined. So it has no derivative at that point.

Answer 7

The question asked me to discuss the continuity in the interval of [1/2, 2]

Answer 8

That's a rational function (ratio of two polynomial functions). It's known that they are continuous everywhere except where the bottom of the ratio is equal to zero, which is at x=0. So on [1/2, 2] it's continuous everywhere.

Answer 9

Okay so rational functions are all continuous except if the bottom of the ratio is zero. Noted.

Answer 10

So how can I understand if a function is continuous without sketching it or graphing it exactly?

Answer 11

You kind of just look at it and try to imagine any problem areas that might come up for certain values of x... That's what I do anyway.

Answer 12

For rational functions, you know you can't divide by zero, so you're going to have a problem when it's zero.

Usually most functions (at least in earlier maths) will just be combinations of a few of the standard functions, where you get some intuitive idea about what could go wrong.

Answer 13

Okay, well I've got a good understanding of it now to complete my homework.

Answer 14

Thank you for your help! I appreciate it! c: