Piecewise Function: Finding Constants for Continuity

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SUMMARY

The discussion centers on the continuity of a piecewise function defined as f(x) = ax^2 + bx + c for -∞ < x < 0, d for x = 0, and (x^2)(sin(1/x)) - 2 for 0 < x < ∞. To ensure continuity across the entire domain, the constants must satisfy c = d = -2. The values of a and b can be any real numbers, as they do not affect the continuity at x = 0.

PREREQUISITES
  • Understanding of piecewise functions
  • Knowledge of limits and continuity in calculus
  • Familiarity with real numbers and their properties
  • Basic trigonometric functions and their behavior
NEXT STEPS
  • Study the concept of limits in piecewise functions
  • Explore the definition of continuity in calculus
  • Learn about the behavior of sin(1/x) as x approaches 0
  • Investigate the implications of choosing different values for constants in piecewise functions
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Students and educators in calculus, mathematicians analyzing piecewise functions, and anyone interested in the principles of continuity in mathematical analysis.

lomantak
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Hello,

Here is a piecewise function that I came over, and it does not seem to have a definite answer, and so I beg of your recondite knowledge to guide me on this one:f(x) =
ax^2 + bx + c if -oo < x < 0
d if x = 0
[(x^2)(sin(1/x))]-2 if 0 < x < oo

Find all values of the constants a, b, c and d that make the function f continuous on -oo < x < oo.

I think a, b and c are all reals, and d is -2, but I am not sure...
 
Last edited:
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For continuity you need c=d=-2. a and b could be anything.
 

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