NDA 2018 Maths · Mathematics · Derivatives · medium

The sum of a function continuous at x=a and a function discontinuous at x=a is discontinuous at x=a. 2. The product of two functions discontinuous at…

Single correct option

Consider the following statements:
1. The sum of a function continuous at

x=a

and a function discontinuous at

x=a

is discontinuous at

x=a

.
2. The product of two functions discontinuous at

x=a

is always discontinuous at

x=a

.
Which of the statements given above is/are correct?

Show answer and solution

Answer: (A) 1 only

Statement 1: if

f+g

were continuous with

f

continuous, then

g=(f+g)-f

would be continuous, a contradiction, so it is true. Statement 2 is false: take

f=g=\operatorname{sgn}(x)

0

; each is discontinuous but

fg=1

for

x\ne0

can be redefined continuous (e.g.

f(x)=1

for

x\ge0

-1

for

x<0

, product

\equiv1

). Hence 1 only.

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