Find all points of discontinuity of f, where f is defined by f(x) = {x^2-3 if x <= 2, x^2+1 if x>2
Answers
The given function is f(x)={
x+1,ifx≥1
x
2
+1,ifx<1
The given function is defined at all the points of the real line.
Let c be a point on the real line.
Case I: c<1, then f(c)=c
2
+1 and
x→c
lim
f(x)=
x→c
lim
(x
2
+1)=c
2
+1
∴
x→c
lim
f(x)=f(c)
Therefore, f is continuous at all points x, such that x<1
Case II : c=1, then f(c)=f(1)=1+1=2
The left hand limit of f at x=1is,
x→1
lim
f(x)=
x→1
lim
(x
2
+1 ) = 1
2
+1=2
The right hand limit of f at x=1 is,
x→1
lim
f(x)=
x→1
lim
(x+1)=1+1=2
∴
x→1
lim
f(x)=f(1)
Therefore, f is continuous at x=1
Case III : c>1, then f(c)=c+1
x→c
lim
f(x)=
x→c
lim
(x+1)=c+1
∴
x→c
lim
f(x)=f(c)
Therefore, f is continuous at all points x, such that x>1
Hence, the given function f has no point of discontinuity.
Step-by-step explanation: