A lemma about upper derivatives (upper, non right)

[Castilla]

Let be

lim sup {g(y) - g(a)} / (y - a) = L.
y-> a

It seems that this lemma exists: For all epsilon e > 0 and delta d > 0, there is an y such that 0 < l y - a l < d and g(y) > L - e.

Can someone give me some hint to proof this?

Thank you.

 

[Castilla]

If someone is interested, there was a mistake in my statement of the lemma.

Le h(y) be = { g(y) - g(a) } / (y - a). What the lemma says is that for any pair of e and d, there is an "y" such that
0 < | y - a | < d and h(y) > L - e (erroneously I put g(y) instead of h(y) in the original statement).

The proof is this (excuse the clumsiness, I can't work on Latex, not for lazyness but for some technical reason beyond my powers):

Lets take and e and d. The known fact is that

limsup h(y) = L.
y->a

So, by definition

L =

lim sup {h(y) / 0 < | y - a | < k }
k->0

So for the selected e there exists a "k" such that k < d and
| sup {h(y) / 0 < |y-a|< k } - L | < e, and therefore
sup {h(y) / 0 < |y-a|< k} > L - e.

But this (and taking on account the definition of supremum) implies that there must be an y such that 0<|y-a|<k (hence 0<|y-a|<d) and h(y) > L - e.

That is all.

 

[tacman]

The lemma about upper derivatives is a statement that relates the limit supremum of a function to its upper derivative. It can be stated as follows:

Let g be a function defined on an interval containing a point a and let L be a real number. If the upper derivative of g at a is equal to L, then for any ε > 0 and δ > 0, there exists a point y such that 0 < |y - a| < δ and g(y) > L - ε.

To prove this lemma, we can use the definition of the upper derivative at a point. By definition, the upper derivative of g at a is given by:
d^*g(a) = lim sup {g(y) - g(a)}/(y - a) as y → a

Let us consider the interval (a - δ, a + δ) where δ > 0. By the definition of the limit supremum, there exists a point y in this interval such that:
g(y) - g(a) > d^*g(a) - ε
or, g(y) > g(a) + d^*g(a) - ε

Since d^*g(a) = lim sup {g(y) - g(a)}/(y - a), it follows that there exists a point z in the interval (a - δ, a + δ) such that:
g(z) - g(a) > d^*g(a) - ε
or, g(z) > g(a) + d^*g(a) - ε

Now, let y be the closer of the two points y and z to a. Since both y and z are in the interval (a - δ, a + δ), it follows that 0 < |y - a| < δ. Also, from the above inequality, we have:
g(y) > g(a) + d^*g(a) - ε = L - ε

Thus, we have found a point y such that 0 < |y - a| < δ and g(y) > L - ε, which proves the lemma.

In conclusion, the lemma about upper derivatives states that for any given ε > 0 and δ > 0, there exists a point y in the interval (a - δ, a + δ) such that 0 < |y - a| < δ and g(y) > L - ε. This lemma