Exercise 3. 1) Let (X, d) be a metric space. Show that f: X → R is Lipschitz with Lipschitz constant Lif and only if f(x) ≤ f(y) + Ld(x, y), for all x, y € X. 2) Suppose again that (X, d) be a metric space. For f₁, the functions max 1 fi, min? 1 fi : X → R are defined by n (min fi)(x)= min (fı(x) ..., fn(x)) and (max fi)(x): = max(fi(x) If each fi, i = 1, ..., n is Lipschitz, show that both min-1 fi and max=1 fi are Lipschitz 3) Show that the functions max : R → R, (x₁,...,n) → max(x₁,...,xn) and min: R" → R, (x₁,...,n) → min(x₁,...,xn) are continuous. 4) If X is a topological space and f₁,..., fn XR are continuous. Show that both min-1 fi and max 1 fi, defined by the equalities (1), are continuous. fn X → Ra finite family of functions, +∞o inf fi(t) = inf f(x), iEN i=0 is finite everywhere, but not continuous. fn(x)), for x = X. (1) 5) Find an example of a countable family of continuous functions fi: [0, 1] :-> [0+ ∞[, i € N, such that the function infien fi : [0, 1] → [0, +∞[, defined by
Exercise 3. 1) Let (X, d) be a metric space. Show that f: X → R is Lipschitz with Lipschitz constant Lif and only if f(x) ≤ f(y) + Ld(x, y), for all x, y € X. 2) Suppose again that (X, d) be a metric space. For f₁, the functions max 1 fi, min? 1 fi : X → R are defined by n (min fi)(x)= min (fı(x) ..., fn(x)) and (max fi)(x): = max(fi(x) If each fi, i = 1, ..., n is Lipschitz, show that both min-1 fi and max=1 fi are Lipschitz 3) Show that the functions max : R → R, (x₁,...,n) → max(x₁,...,xn) and min: R" → R, (x₁,...,n) → min(x₁,...,xn) are continuous. 4) If X is a topological space and f₁,..., fn XR are continuous. Show that both min-1 fi and max 1 fi, defined by the equalities (1), are continuous. fn X → Ra finite family of functions, +∞o inf fi(t) = inf f(x), iEN i=0 is finite everywhere, but not continuous. fn(x)), for x = X. (1) 5) Find an example of a countable family of continuous functions fi: [0, 1] :-> [0+ ∞[, i € N, such that the function infien fi : [0, 1] → [0, +∞[, defined by
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Exercise 3
Need part 1, 2, 3, 4 and part 5
![Exercise 3. 1) Let (X, d) be a metric space. Show that f: X → R is Lipschitz with Lipschitz
constant L if and only if
f(x) ≤ f(y) + Ld(x, y), for all x, y ≤ X.
2) Suppose again that (X, d) be a metric space. For f₁, fn X→ Ra finite family of functions,
the functions max 1 fi, min-1 fi: X → R are defined by
n
(min fi)(x) = min(f₁(x)….., fn(x)) and (max fi)(x) = max(ƒ₁(x)…..,‚ ƒn(x)), for x € X. (1)
If each fi, i = 1,...,n is Lipschitz, show that both min-1 fi and max 1 fi are Lipschitz
3) Show that the functions max : R → R, (x₁,...,xn) → max(x₁,...,xn) and min: R" →
R, (x₁,...,xn) → min(x₁,...,xn) are continuous.
4) If X is a topological space and f₁,..., fn X → R are continuous. Show that both min-1 fi
and max 1 fi, defined by the equalities (1), are continuous.
5) Find an example of a countable family of continuous functions fi: [0, 1] → [0 + ∞[, i € N,
such that the function infien fi : [0, 1] → [0, +∞[, defined by
inf fi(t)
ŻEN
is finite everywhere, but not continuous.
=
+∞o
inf fi(x),
i=0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa68164dd-6bba-4aa5-92bc-4824a71db092%2F7ef2bf05-f8fb-4c70-b86e-8a15c47023b7%2Fnze6ma_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Exercise 3. 1) Let (X, d) be a metric space. Show that f: X → R is Lipschitz with Lipschitz
constant L if and only if
f(x) ≤ f(y) + Ld(x, y), for all x, y ≤ X.
2) Suppose again that (X, d) be a metric space. For f₁, fn X→ Ra finite family of functions,
the functions max 1 fi, min-1 fi: X → R are defined by
n
(min fi)(x) = min(f₁(x)….., fn(x)) and (max fi)(x) = max(ƒ₁(x)…..,‚ ƒn(x)), for x € X. (1)
If each fi, i = 1,...,n is Lipschitz, show that both min-1 fi and max 1 fi are Lipschitz
3) Show that the functions max : R → R, (x₁,...,xn) → max(x₁,...,xn) and min: R" →
R, (x₁,...,xn) → min(x₁,...,xn) are continuous.
4) If X is a topological space and f₁,..., fn X → R are continuous. Show that both min-1 fi
and max 1 fi, defined by the equalities (1), are continuous.
5) Find an example of a countable family of continuous functions fi: [0, 1] → [0 + ∞[, i € N,
such that the function infien fi : [0, 1] → [0, +∞[, defined by
inf fi(t)
ŻEN
is finite everywhere, but not continuous.
=
+∞o
inf fi(x),
i=0
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VIEWStep 3: (2) Showing that both max(f) and min(f) are lipschitz
VIEWStep 4: (3) Proving the continuity of the given functions
VIEWStep 5: (4) Proving the continuity on the topology X
VIEWStep 6: (5) Finding countable family of continuous functions
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