Question

Find the values of the constants a and b for which the following function is continuous, but not differentiable.


\[
f(n) = \left\{
\begin{array}{l l}
ax+b, & \quad x>0;\\
\sin2x & \quad x \le 0.\\
\end{array} \right\}
\]

Answer 1

Following the theorem of diffentiation; if a function is differentiable, it is continuous, so how am I going to do this?

Answer 2

@asnaseer Help please :D

Answer 3

@myininaya Help!

Answer 4

@radar

Answer 5

The only help that I can offer is, I hope myininaya gets over here.

Answer 6

Ahh. D:

Answer 7

If ƒ is differentiable at a point x, then ƒ must also be continuous at x.
see: en.wikipedia.org/wiki/Differentiable_function

Answer 8

are you sure you have stated the question correctly?

Answer 9

Yes, it's an MIT worked example. http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/part-a-definition-and-basic-rules/session-4-limits-and-continuity/MIT18_01SCF10_ex04prb.pdf

Answer 10

hmmm.... maybe the fact that f(x) = ax +b for x > 0 makes it non-differentiable at x=0?

Answer 11

the way I would tackle this in that case is as follows...

Answer 12

find f(0)

Answer 13

what do you get?

Answer 14

Hold on

Answer 15

b?

Answer 16

look carefully at the definition of f(x)

Answer 17

at x=0:\[f(x)=\sin(2x)\]

Answer 18

so f(0) = ?

Answer 19

0?

Answer 20

correct

Answer 21

now, next we look at the limit as x tends to \(0^+\)

Answer 22

what would that give?

Answer 23

i.e. what is the limit of f(x) as x tends to zero from the positive side of the x-axes?

Answer 24

remember on the positive x-axes, f(x) = ax + b

Answer 25

So we take the \[\large \lim_{ x\rightarrow 0 } \sin(2x)\]?

Answer 26

no - look at what I just typed ^^^

Answer 27

I'm confused. :|

Answer 28

what is the f(x) defined as for x>0?

Answer 29

ax+b?

Answer 30

correct, so we know on the positive side of the x-axes, f(x) = ax + b
so what does this tend to as x tends to 0 (from the positive side)?

Answer 31

do you understand zepp?

Answer 32

x tends to 0+ is equal to x tends to 0-? Since the ax+b doesn't lie on the y-axis?

Answer 33

@arghya - please let zepp answer this

Answer 34

let me draw a diagram to help here