Proving Continuity of g(x) at x=3: Using Real Definition and Limit Method

  • Thread starter Thread starter Math_Geek
  • Start date Start date
Math_Geek
Messages
23
Reaction score
0
HELP! I do not understand!

Homework Statement


Define f(x)=2x, x is rational and x+3 when x is irrational. Find all points where g(x) is continuous and prove continuity at these points


Homework Equations


From analysis homework and using the real definition of continuity


The Attempt at a Solution



Graphing g(x) I can see that the only point that will be continuous will be at x=3, how would you show using the definition that this point is continuous. Also is this the only point. I have been struggling, please help a girl in distress. lol
 
Physics news on Phys.org
… 'tis I … tiny-tim! …

Math_Geek said:
Graphing g(x) I can see that the only point that will be continuous will be at x=3, how would you show using the definition that this point is continuous. Also is this the only point. I have been struggling, please help a girl in distress. lol

:smile: … don't worry that pretty little head … :smile:
​

Just choose an epsilon, and find a delta such that f(3 + x) - 6 < epsilon, for all x < delta, and all your worries will go away! :smile:
 
so I just solve it like a limit problem? Thank you so much, I just knew someone would help a girl in distress lol.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
View full post »

Similar threads

f(x) = 2x+1, proving that it is continuous when p = 1 with 𝛿 and ε
Replies
7
Views
1K
Can this function still be a constant function?
Replies
11
Views
1K
Is the given function continuous at x= 0? f(x) = {sin(1/x) if x≠0, 1
Replies
4
Views
4K
Proving Irregularity of {(x, y, z) within R^3 : x^2 + y^2 - z^2 = 0}
Replies
14
Views
3K
Prove ##(a+b) + c = a + (b+c)## using Peano postulates
Replies
9
Views
2K
A few questions to make sure I understand Fourier series
Replies
3
Views
2K
Prove limit comparison test for Integrals
Replies
2
Views
2K
Medium Hard continuity proof Tutorial Q6
Replies
2
Views
1K
Prove ##(a\cdot b)\cdot c =a\cdot (b \cdot c)## using Peano postulates
Replies
1
Views
1K
Understanding the solution to a calculus problem (removable discontinuities)
Replies
2
Views
2K

Hot Threads

Recent Insights

Back
Top