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Continuity quiz

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  • What is the definition of continuity at a point c for a function?
    A function is continuous at c if the limit as x approaches c equals the function value at c.
  • How can you visually check continuity at a point on a graph?
    You can trace through the function at the point without lifting your pen; if you don't have to stop, the function is continuous there.
  • What are the three main types of discontinuities seen in functions?
    Discontinuities can occur as holes, jumps, or asymptotes.
  • Where do discontinuities commonly occur in rational functions?
    Discontinuities often occur where the denominator equals zero, which may indicate holes or asymptotes.
  • How do you determine continuity at a specific point using limits and function values?
    Compare the limit as x approaches the point and the function value; if they are equal, the function is continuous at that point.
  • What happens if the limit as x approaches c does not exist for a function?
    If the limit does not exist, the function is discontinuous at c.
  • What type of discontinuity occurs when the function value is undefined at a point but the limit exists?
    This is called a hole or removable discontinuity.
  • How do you find where a rational function is discontinuous?
    Set the denominator equal to zero and solve for x; these values are where the function is discontinuous.
  • What is a jump discontinuity in a piecewise function?
    A jump discontinuity occurs when the function 'jumps' from one value to another at the boundary between pieces, and the limits from each side are not equal.
  • How do you check for discontinuity at the boundary of a piecewise function?
    Find the left and right-sided limits at the boundary and compare them to the function value; if they are not all equal, there is a discontinuity.
  • What is the function value and limit for f(x) at x = 2 if both are 4?
    The function is continuous at x = 2 because the limit and function value are both 4.
  • What does it mean if you have to pick up your pen when tracing a graph at a point?
    It means the function is discontinuous at that point.
  • What causes discontinuity at x = 4 in a piecewise function where left and right limits are not equal?
    A jump discontinuity occurs because the limits from each side are different.
  • What is the discontinuity at x = 1 if the function approaches negative infinity from both sides?
    There is an asymptote at x = 1, causing the function to be discontinuous there.
  • How do you determine if a piecewise function is continuous at the point where the pieces meet?
    Check if the left and right-sided limits and the function value at the meeting point are all equal; if not, the function is discontinuous there.