caledonia

It is defined at x = 1/2, namely f(1/2) = 0. I used 0.5 instead of the symbol 1/2 !

I was wrong in my original post - it is not true that a function which maps closed (sub) intervals to closed (sub) intervals is necessarily continuous. Here is a counter example: Define f(x) as sin(1/(0.5-x)) for 0 <= x < 0.5, f(0.5) = 0, f(x) = sin(1/(x-0.5)) for 0.5 < x <= 1. Then f is discontinuous at 0.5, because for all delta > 0, there exists points nearer to 0.5 whose values are 1. But the image of any subinterval containing the point 0.5 is [-1,1] ; and the image of any subinterval not containing 0.5 is also a closed interval, because the function is here continuous and the intermediate value theorem applies.

No I don't. Nor can I find a counter-example for closed intervals. Hence unknown whether my assertion is true or not !

From the usual epsilon, delta definition of continuity, two significant theorems follow : (i) a function continuous on a closed interval attains in supremum and infimum on that interval ; and (ii) for any two values in the range of the function, any intermediate value also belongs to the range. From these theorems, one can deduce that the image of any sub-interval of the domain is a sub-interval of the range. And it turns out that this is a necessary and sufficient condition for continuity. So we can use this last as the very definition of continuity for a function on a closed interval. Some students, who find the conventional definition of 'continuity at a point' disconcerting, may find this preferable.

You do not explicitly mention the loss of species which is happening now and is accelerating, owing to man's "massive scale" impact on the environment. For me, this is the most deplorable result of unrestrained human reproduction.

Er, yes please. But let me explain further my problem. I read that Einstein in his tramcar, heading away from the town clock in Berne, considered what would happen if his speed approached that of light. He deduced that the clock would appear to stop. I can 'understand' this, by assuming that he would be 'riding', i.e. keeping pace with, the wavefront. But if I reverse this situation and try to apply similar reasoning, I find that the clock is then seen to be moving at double speed. Whereas SR says infinite speed. How does the model change when taking into account the basic postulate (for light) of SR ?

Thank you. Can you explain these formulae in terms of the basic postulate viz. the constancy of the speed of light ?

With my telescope I can see a distant clock, keeping 'normal' time. (taking both time dilation and doppler into account), What do I SEE if I move away from the clock at high speed, say 3/4 speed of light ? What do I see if my speed approaches c ? On the other hand, what do I SEE if I move quickly, say 3/4 c, towards the clock ? What do I see if my speed approaches the speed of light ? Four questions . . . Thank you.

m^2 = n^2 * 2 has no solution for integers m and n because root two is irrational. But m^2 = n^2 * 2 -1 does have solutions, the first of these being 7&5, 41&29, 239&169, 1393&985, 8119&5741. I believe that there are an infinite number of solutions, in other words for all N, there exists a solution with m and n both greater than N. Can anyone give me a proof ?

In Lemma XIII of the Principia, Newton says "The latus rectum of a parabola belonging to any vertex is quadruple the distance of that vertex from the focus of the figure. And this is demonstrated by the writers on the conic sections". Well, Apollonius does not state this proposition. And I can't find any 'writer' who does. Hence, can anyone help ? Thanks.

In the proofs of the inverse square law for planetary orbits, as deduced by Newton in the Principia from Keplers Laws, he assumes lots of properties of the conics. Mathematicians in his day were well versed in Geometry, including the Conics. Not so nowadays. I started with Apollonius but found it tough going - many proofs are very long. Archimedes showed that an ellipse is also a section of a right cylinder, and this simplifies some theorems. Then again, Dandelin spheres are great for proving other properties. I found two other proofs myself, much easier than Apollonius, in the case of the parabola. If anyone else would like to be able to follow Newton's geometrical proofs, I could post my documents (as a set of pdf's).

Does DNA profiling measure two sequences , one from each homologous chromosome ? How is the information used ? thank you

This and similar propositions are easy to prove once one has a good definition of real numbers - see www.realnumbers.me.uk for example

many thanks for clear explanation

If I travel at high speed towards a clock, do I see it as running fast? If yes, how does this square with the assertion of Time Dilation viz. "each observer sees the other's clock as running slow".