Question

find all points of discontinuity of F , where F is defined by F(x)={x^2+1,if x <1. and x+1,if x less than or equal to 1​

Answer 1

Answer:

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. and please make me Brainliest