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