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Theorems of Continuity

  1. Sum, difference, product, and quotient of two continuous functions is always a continuous function. However, h(x) = f(x)/g(x) is continuous at x = a only if g(a) ≠ 0.
  2. If f(x) is continuous and g(x) is discontinuous then f(x) + g(x) is a discontinuous function.
     
    Illustration If f(x) = x and g(x) = [x], greatest integer function.
     
    Sol. Here f(x) is continuous at x = 0 but g(x) is discontinuous at x = 0.
     
    Hence F(x) = x + [x] is discontinuous at x = 0 as f(0+) = 0 and f(0) = –1.
  3. If f(x) is continuous and g(x) is discontinuous at x = a then the product function h(x) = f(x) g(x) is not necessarily be discontinuous at x = a.
     
    Illustration Consider functions f(x) = x3 and g(x) = sgn(x).
     
    Sol. Here f(x) is continuous at x = 0 and g(x) is discontinuous at x = 0.
     
    But product function F(x) = f(x) g(x) = 88088.png is continuous at x = 0.
  4. If f(x) and g(x) are discontinuous at the same point then sum or product of the functions may be continuous. For example, f(x) = [x] (GIF function) and g(x) = {x} (fractional part function) both are discontinuous at x = 1 but their sum f(x) + g(x) = x is continuous at x = 1.
     
    f(x) = 88082.png and g(x) = 88076.png
     
    Here both the functions are discontinuous at x = 0, but their product f(x) × g(x) = –1, x R is continuous at x = 0.
  5. Every polynomial is continuous at every point of the real line.
  6. Every rational function is continuous at every point where its denominator is different from zero.
  7. Logarithmic functions, exponential functions, trigonometric functions, inverse circular functions, and modulus functions are continuous in their domain.




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