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Math Help - continuity & differentiability, limits

  1. #1
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    Unhappy continuity & differentiability, limits

    Give an example of a function, if such function exists (by sketching OR writing it) that satisfies the conditions given in a), b), c) (each separately):

    a) f(-2)=4, f is continuous at x=(-2) but has no derivative at this point

    b) f is not defined at x=(-3) but has a limit at x=(-3) and is not continuous at x=(-3)

    c) f is continuous at x=(-4) from the right and from the left but is not "continuous" at this point
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  2. #2
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    Quote Originally Posted by calculus_2718 View Post
    Give an example of a function, if such function exists (by sketching OR writing it) that satisfies the conditions given in a), b), c) (each separately):

    a) f(-2)=4, f is continuous at x=(-2) but has no derivative at this point

    b) f is not defined at x=(-3) but has a limit at x=(-3) and is not continuous at x=(-3)

    c) f is continuous at x=(-4) from the right and from the left but is not "continuous" at this point
    I can give you examples for the questions a) and b) but not for c):

    a) f(x)=-|x+2|+4 See the first graph.


    b) f(x) = e^{-\frac1{x+3}} See second graph. ( The exponent is -\dfrac1{x+3}

    You can prove that \lim_{x \to -3} f(x) = 0

    As I've mentioned above I can't think of an example which satisfies the third question.
    Attached Thumbnails Attached Thumbnails continuity & differentiability, limits-betrg_nondiffbar.png   continuity & differentiability, limits-nodef_mitlim.png  
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  3. #3
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    Quote Originally Posted by calculus_2718 View Post
    Give an example of a function, if such function exists (by sketching OR writing it) that satisfies the conditions given in a), b), c) (each separately):

    a) f(-2)=4, f is continuous at x=(-2) but has no derivative at this point

    b) f is not defined at x=(-3) but has a limit at x=(-3) and is not continuous at x=(-3)

    c) f is continuous at x=(-4) from the right and from the left but is not "continuous" at this point
    Some examples
    (i) y = \sqrt{x+2} + 4

    (ii) y = \frac{x^2-9}{x+3}

    So now the question is why?
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  4. #4
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    Quote Originally Posted by calculus_2718 View Post
    Give an example of a function, if such function exists (by sketching OR writing it) that satisfies the conditions given in a), b), c) (each separately):

    a) f(-2)=4, f is continuous at x=(-2) but has no derivative at this point
    f(x)= |x+2|+ 4 works.

    b) f is not defined at x=(-3) but has a limit at x=(-3) and is not continuous at x=(-3)
    f(x)= 1 for all x except -3 and not defined at x= -3. If you prefer a "formula", f(x)= \frac{x+3}{x+3} or [tex]f(x)= \frac{x^2- 9}{x+ 3} will work.

    c) f is continuous at x=(-4) from the right and from the left but is not "continuous" at this point
    This can't be done. "continuous at x= -4 from the right" means \lim_{x\rightarrow -4^+} f(x)= f(-4) and "continuous at x= -4 from the left" means \lim_{x\rightarrow -4^-} f(x)= f(-4). Those both say that f(-4) exists and together they say \lim_{x\rightarrow -4} f(x) exists and is equal to f(-4), precisely the condition that f be continuous at x= -4.
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