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Real Analysis
Let I:=[a,b] and let f: I→R be differentiable at c ∈ I. Show that for every ϵ >0 there exists δ>0 such that if 0<|x-y|<δ and a ≤x ≤c ≤y ≤b, then |[((f(x)-f(y))/(x-y)] -f '(c)|<ϵ.
Real Analysis
Continuity:
Let the function f(x) be continuous at every point of a closed interval [a,b]. Assume that f(x) > 2 for every x in [a,b]. Prove that there exists number c > 2 such that f(x) > c for every x in [a,b].
Let the function f(x) be continuous at every point of a closed interval [a,b]. Assume that f(x) > 2 for every x in [a,b]. Prove that there exists number c > 2 such that f(x) > c for every x in [a,b].
Real Analysis
Uniform continuity:
Prove that if f and g are uniformly continuous on R and are bound, then f*g is uniformly continuous on R.
Prove that if f and g are uniformly continuous on R and are bound, then f*g is uniformly continuous on R.
Real Analysis
if f(x)=x and g(x)=(1/x), find (f/g)(x)
Real Analysis
how to prove this equation A∩B⊂A⊂A⊂B
Real Analysis
solve series ∑(2n-1)/(n(n+1)(n+2)), where n=1,2,...,∞
Real Analysis
What are the points of continuity and the points of discontinuity of the following function? Justify your answer.
f(x) = x if x is rational.
f(x) = x^3 if x is irrational.
f(x) = x if x is rational.
f(x) = x^3 if x is irrational.
Real Analysis
Let the function f(x) be continuous at every point of a closed interval [a,b]. Assume that f(x) > 2 for every element x in [a,b]. Prove that there exists a number c>2 such that f(x) > c for every element x in [a,b].
Real Analysis
Let the function f(x) be continuous at a point X_o. Assume that f(X_o) > 2. Prove that there exists a neighborhood of X_o such that f(x) > 2 for every x in this neighborhood.
Real Analysis
Prove (using the "epsilon - delta" definition) that the function f(x) = 4x^2 - 5x + 3 is continuous at every point X_0.
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#332078 on Oct 2022
#332078 on Oct 2022