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Home Continuous Functions Continuity Theorem 2:
See also: Limit Rules 
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Theorem 2:

THEOREM 2

Suppose f and g are continuous at c.

(i)     For any constant k, the function k · f(x) is continuous at c.

(ii)    f(x) + g(x) is continuous at c.

(iii)    f(x) · g(x) is continuous at c.

(iv)    If g(c) ≠ 0, then f(x)/g(x) is continuous at c.

(v)    If f(c) is positive and n is an integer, then 03_continuous_functions-102.gif is continuous at c.

By repeated use of Theorem 2, we see that all of the following functions are continuous at c.

  • Every polynomial function.
  • Every rational function f(x)/g(x), where f(x) and g(x) are polynomials and g(c) ≠ 0.
  • The functions f(x) = xr, r rational and x positive.
  • Sometimes a function f(x) will be undefined at a point x = c while the limit
    L = limx→c f(x)
    exists. When this happens, we can make the function continuous at c by defining f(c) = L.
Home Continuous Functions Continuity Theorem 2:

Last Update: 2006-11-06