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Unformatted text preview: R, and let I be
a closed interval in R. Prove that the set f −1 (I ) is a closed set, where
f −1 (I ) = {x ǫ R2 : f (x) ǫ I } .
Indicate clearly where you are using that f (x) is continuous. (Suggestion: in your argument, use 25.2,(1c) as the deﬁnition of cluster point.) 4. (15: 7,3,5) The convolution f (x) ∗ g (x) of two continuous functions f (x) and g (x) is a
function of x deﬁned by the formula
�x
f (x) ∗ g (x) =
f (t) g (x − t) dt
0 . d
f ∗ g ; the formula should be expressed
dx
′
′
only in terms of f (x), g (x), f (x), g (x), f (0), g (0) and their convolutions.
a) Find, with proof, a formula for its derivative eax − ebx
.
a − b (Note that the convolution is linear: (cf ) ∗ g = c(f ∗ g ). eax ∗ ebx = b) Test your formula on the convolution c) Using your formula, prove that y = f (x) ∗ sin x is a solution to y ′′ + y = f (x) on
(−∞, ∞)
� ∞ −xt
e
5. (15)
Let f (x) =
dt .
1+t
0
Prove that f (x) is deﬁned and diﬀerentiable for x > 0, and is a solution to the diﬀerential
equation y ′ − y = 1/x, x > 0. 1 6. (15: 8,7)
(a) Prove the set M of all square matrices with integer entries is countable.
(b) Let...
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This note was uploaded on 01/22/2014 for the course MATH 18.100A taught by Professor Arthurmattuck during the Fall '12 term at MIT.
 Fall '12
 ArthurMattuck

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