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Math Help - Finding where a function is continuous

  1. #1
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    Finding where a function is continuous

    The question:

    Determine at which points the function f: R -> R is continuous.

       f(x) = \left\{<br />
     \begin{array}{lr}<br />
       e^{2x} &x < 0 \\<br />
       cos x   &x \geq 0<br />
     \end{array}<br />
   \right.

    I'm not sure how to do this. I know how to check if a function is continuous at a given point, but I don't know how to find those points. Is it just a matter of looking at it, and determining that the exponential and cosine functions are continuous for all x, therefore the piecewise function is also continuous for all x?
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  2. #2
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    Quote Originally Posted by Glitch View Post
    The question:

    Determine at which points the function f: R -> R is continuous.

     f(x) = \left\{<br />
\begin{array}{lr}<br />
e^{2x} &x < 0 \\<br />
cos x &x \geq 0<br />
\end{array}<br />
\right.

    I'm not sure how to do this. I know how to check if a function is continuous at a given point, but I don't know how to find those points. Is it just a matter of looking at it, and determining that the exponential and cosine functions are continuous for all x, therefore the piecewise function is also continuous for all x?
    The potential problem with continuity will occur at x = 0 (why?). Try drawing the graph of f(x). What happens at x = 0? Draw a conclusion. (The argument can be made rigorous using limits and the definition of continuity).
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  3. #3
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    So I should check for continuity at 0 by finding the limit and the function value at 0, then see if they're equal? (I believe that's the method my textbook states). Drawing the graph would look like a 'V', with vertex at 0, if I remember correctly.
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  4. #4
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    Quote Originally Posted by Glitch View Post
    So I should check for continuity at 0 by finding the limit and the function value at 0, then see if they're equal? (I believe that's the method my textbook states). Drawing the graph would look like a 'V', with vertex at 0, if I remember correctly.
    Yes. The left hand and right hand limits have to be equal. In fact, they have to both be equal to f(0).
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  5. #5
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    Thank you.
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