Question

if f and g are 2 real function continuos at a real number c then f+g is continous at x=c ?? explain how?​

Answer 1

Given: f and g are two real functions continue on all real numbers

To find: continuity of (f+g),(fg),

g

f

andf(g(x))atx=c

Sol: lim

x→c

−

f(x)=lim

x→c

−

f(x)=f(c)

Similarly lim

x→c

−

g(x)=lim

x→c

+

g(x)=g(c)

We'll look for options now

(f+g)

lim

x→c

−

{f(x)+g(x)}=lim

x→c

−

f(x)+lim

x→c

−

g(x)=f(c)+g(c)

lim

x→c

+

{f(x)+g(x)}=lim

x→c

+

f(x)+lim

x→c

+

g(x)=f(c)+g(c)

LHL=RHL⟹(f+g) is continuous

f.g

lim

x→c

−

f(x).g(x)=f(c).g(c)=lim

x→c

+

f(x).g(x)

⟹f.g is continuous

lim

x→c

−

f(g(x))=f(g(c))=lim

x→c

+

f(g(x))

⟹fg is continuous

g

f

lim

x→c

−

g(x)

f(x)

=lim

x→c

+

g(x)

f(x)

=

g(c)

f(c)

⟹

g

f

is continous for all CϵR except when g(c)=0

Hence, (f+g),fg,f.g and

g

f

all are continuous for all CεR

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