Find all the points of discontinuity of the function
f
defined by
f(x)= {
x+2, if x<10, if x=1x−2, if x>1
Get the complete, step-by-step math solution for: "Find all the points of discontinuity of the function f defined by f(x)= \{ {rr} x+2, & { if } x<1 0, & { if } x=1 x-2, & { if } x>1". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Analyze continuity for
x<1
For all values of
x<1
, the function is defined as
f(x)=x+2
. This is a polynomial function, and polynomial functions are continuous everywhere. Therefore, f(x) is continuous for all
x<1
.
Step 2: Analyze continuity for
x>1
For all values of
x>1
, the function is defined as
f(x)=x−2
. This is also a polynomial function, and thus it is continuous for all
x>1
.
Step 3: Check continuity at
x=1
To check for continuity at
x=1
, we need to evaluate the left-hand limit, the right-hand limit, and the function value at
x=1
. The left-hand limit is
x→1−lim(x+2)=3
. The right-hand limit is
x→1+lim(x−2)=−1
. The function value at
x=1
is given as
f(1)=0
.
Step 4: Determine discontinuity at
x=1
Since the left-hand limit (3) is not equal to the right-hand limit (-1), the limit of the function as
x
approaches 1 does not exist. For a function to be continuous at a point, the limit must exist and be equal to the function's value at that point. Therefore, the function is discontinuous at
x=1
.