Continuity of a function module02
- 1. CONTINUITY OF A FUNCTION AT A NUMBER A function is continuous at a number , if and only if the following conditions are satisfied: 1) is in the domain of function 2) A function is discontinuous at a number , if one of the conditions is not satisfied. EXAMPLE #1: Verify that is continuous at . EXAMPLE #2: Verify that is continuous at . CONTINUITY OF A FUNCTION ON AN INTERVAL 𝑓 𝑥 3𝑥 1 𝑖𝑓 𝑥 1 𝑓 1 2 1 𝑓 1 1 𝑓 𝑥 𝑥2 1 𝑖𝑓 𝑥 1 𝑓 1 1 1 𝑓 1 2 1st Equation 𝑓 1 3 1 1 ---------------------- Substitute the 𝑎 1 to the equation. 2nd Equation 𝑓 1 1 2 1 ---------------------- Substitute the 𝑎 1 to the equation. THEREFORE, THE FUNCTION MEETS THE CONDITIONS OF A CONTINUOUS FUNCTION. 𝒙 𝟏 𝟎 𝒙 𝟎 𝟏 𝒙 𝟏 For you to know if the function given is continuous at 𝒂 𝟏 by identifying the domain of the function, since the given is rational function the value of x that should not be included in the domain is 1. SINCE THE FIRST CONDITIONIS NOT SATISFIED THEN THE FUNCTION IS NOT CONTINUOUS AT 𝒂 𝟏
- 2. A function is continuous on , if it is continuous at every point in . EXAMPLE #3: Verify that √ is continuous over the interval of 1 1 For you to know if the function given is continuous at the interval of 1 1 by identifying the domain of the function, since the given is a polynomial function the value of x are all real numbers. SINCE THE CONDITIONS ARE SATISFIED THEN THE FUNCTION IS NOT CONTINUOUS AT HE INTERVAL OF 𝟏 𝟏