Select any statement that is always true: f(x) is...

Select any statement that is always true: f(x) is non-differentiable at any discontinuity. If f(x) is differentiable at x=a, then f(x) is continuous at x=a If f(x) is continuous at x=a, then f(x) is differentiable at x=a f(x) is non-differentiable at any discontinuity

Transcript

00:01 Okay, so we've asked some statements about differentiability and continuity.

00:04 So let's remind ourselves of what they are.

00:07 So continuity basically means, so you can define it technically, but it essentially means there's no break in the function.

00:17 So for example, if we have a function like this, there's no break, so it's continuous.

00:25 If we have a function like this, then there's a break here, it jumps down, and so this is not continuous.

00:39 A function like this, it's got a vertex here, but you can see that it still doesn't break, it still follows itself all the way, you can draw the whole thing without your pen coming off the page, and so that's continuous.

00:51 Whereas differentiability means that it has to be continuous and smooth.

01:04 Now by smooth, we just mean there needs to be a well -defined tangent tangent line at that point.

01:10 So if we draw our graphs again, we can see that this is continuous everywhere and it's smooth, so it's differentiable.

01:18 This graph, we can see there isn't a tangent line at this x value because it's not even continuous.

01:27 The limit as you go to above the x value does not equal the limit as you go to below, and so the tangent line isn't going to to make any sense.

01:37 And then this graph, while it's continuous everywhere, there's no break, we can see that at this point there's a point, there's a vertex, so it's not smooth there, it's not like there's a nice curve, there's a point...

Question

Select any statement that is always true: f(x) is non-differentiable at any discontinuity. If f(x) is differentiable at x=a, then f(x) is continuous at x=a If f(x) is continuous at x=a, then f(x) is differentiable at x=a f(x) is non-differentiable at any discontinuity

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