SUMMARY:
1.If a function f is said to be continuous at a point c if and only if
f(c) = f(x).
thus,three conditions must hold for f to be continuous at c
(a) f must be defined at c
(b) the limit of f at c must exist in .
(c) these two values must be equal.
solution as follows
[4 Pts. Use the definition of continuity to show that the function f is continuous at <=0 10 g(x)= 3-4
(1) Given a continuous function f, show that raC Hint: parts. (1) Given a continuous function f, show that raC Hint: parts.
Show that the function is continuous at point 1, by using f(:0) = +2 4.72 – 3 9-3
1. Consider the function f(a) 2-9 3, but not differentiable at z Show that f( ) is continuous at r- 3.
4. (a) Let A [0, oo) and let f.g:AR be functions which are continuous at 0 and are such that f(0) 9(0)-1. Show that there exists some δ > 0 such that ifTE 0,d) then (b) Consider the function 0 l if z e R is rational, if zER is irrational f(z) Show that limfr) does not exists for any ceR. 4. (a) Let A [0, oo) and let f.g:AR be functions which are continuous at 0 and are such...
Question 9 3 pts The Laplace transform of the piecewise continuous function 4, 0<t <3 f(t) is given by t> 3 (2, L{f} = { (1 – 3e-*), s>0. O 2 L{f} (2 - e-st), 8 >0. 2 L{f} = (3 - e-st), s >0. O None of them 1 L{f} (1 – 2e -st), s >0.
Question 4* (Similar to 18.1) Suppose f is a continuous function on a closed interval [a, b]. In class, we proved that f attains its maximum on that interval, i.e. there exists Imar E la, so that f(Imar) > f(x) for all r E (a,b]. We didn't prove that f attains its minimum on the interval, but I claimed that the proof is similar. In fact, you can use the fact that f attains its maximum on any closed interval...
I. Let f : R → R be a continuous function. Show that ER sup is a Fo set I. Let f : R → R be a continuous function. Show that ER sup is a Fo set
2. Rolle's theorem states that if F : [a, b] → R is a continuous function, differentiable on Ja, bl, and F(a) = F(b) then there exists a cela, b[ such that F"(c) = 0. (a) Suppose g : [a, b] → R is a continuous function, differentiable on ja, bl, with the property that (c) +0 for all cela, b[. Using Rolle's theorem, show that g(a) + g(b). [6 Marks] (b) Now, with g still as in part (a),...
9. Is the function f(x) = sin 1/x continuous on (0,1)? Is it uniformly con- tinuous on (0,1). Justify your answers. 10. Is the function f(x) = x sin 1/x uniformly continuous on (0, 1)? Justify your answer.