When is the function not continous...

[mreiki]

.          ( sin(x)/x , x < a    
    f(x) = (              
.          ( 1         , x >= a

When is f not continous if 1) a=0, 2) a<0, 3)a>0 ?

I got the answers 1) nowhere, 2) when x=a and 3) when x=0 or x=a
But i assumed that sin(x)/x only has the limit 1 when x is heading towards 0, and that otherwise sin(x)/x never takes the value 1.

Is there any simple way to prove that? (that sin(x)/x never takes the value 1

 

[mreiki]

Alright, i think i found the proof as a part of another proof in my math book, but thanks anyways.

 

[pka]

Is there any simple way to prove that sin(x)/x never takes the value 1

Of course is |x|>1 then it clearly true.
We usually just apply the mean value theorem.
The derivative of the sine is the cosine which is less than or equal 1.

 

[mreiki]

I do realize that it's quite obvious if x > 1 or x < -1 ...
Also if -1 < x < 0, then it's quite obvious since Sinx is then positive but x, negative, giving a negative result. But i was in doubt when 0 < x < 1 ...
We haven't learned differentiation yet, so the mean value theorem is not in the picture.

But it can be proven when 0 < x < pi/2 by other means by comparing the areas of a sector OAB, spanned by the angle x, and the triangle OAB that the middle O and the string AB of the sector form. Then, since AREAtriangle < AREAsector, you get 1/2sinx < 1/2*x*1 <=> sinx/x <1