Use the definition of uniform continuity to prove that f(x)is uniformly continuous on , 00
definition of continuity to prove that f : (0,00) by f(x)-13 + 1 is continuous at every Zo 0. Use the є-ð definition ) Use the є- R defined that g(x)-_a_ is continuous at every a є (-1,00) +1
9. Prove that the function f(x) = ax+b is uniformly continuous on R by directly applying the e, 8 definition of uniform continuity.
Let f:D + R be a function. (a) Recall the definition that f is uniformly continuous on D. (You do not need to write this down. This only serves as a hint for next parts.) (b) Use (a) and the mean value theorem to prove f(x) = e-% + sin x is uniformly continuous on (0, +00). (c) Use the negation of (a) to prove f(x) = x2 is not uniformly continuous on (0,0).
Use the definition of continuity to determine whether f is continuous at a x2 - 169 f(x) = if x #13 4 if x = 13 a = 13 The function | at 13 is continuous is not continuous
try to use the definition of uniformly continuous to prove this
question, thank you so much!
3. Determine which of the following continuous functions are uniformly continuous on the given set. Justify your answers. * (a) f(x) =* on [2,5]
[4 Pts. Use the definition of continuity to show that the function f is continuous at <=0 10 g(x)= 3-4
Please help me prove this! This is a real analysis question on
uniform continuity.
Prove the following statement: Proposition 2. If f : (a,c) + R is such that f is uniformly contin- uous on both (a, b] and [b,c) for some b € (a,c), then f is uniformly continuous on (a,c).
Use the sequential criterion for the absence of uniform continuity to show that the function f(0, 1) - 1 given by f(x) = 1/-x) is not uniformly continuous. Theorem 4.4.5 (Sequential Criterion for Absence of Uniform Conti- nuity). A function /: A → R fails to be uniformly continuous on A if and only if there erists a particular to > 0 and two sequences (in) and (y) in A satisfying In - yn — but if(n)-f(n) Co.
Prove that f(x) = is uniformly continuous on (1,00) and not uniformly continuous on (0,1). (19 pts)
7. Using the definition of continuity directly to prove that f: (1,00) + R defined by f(3) = and f(1) = 0 is continuous on at 2 = 2 but not continuous at 1.