We have
lim(
)( )
x
c f
g
x
→
+
 = lim[ ( )
( )]
x
c f x
g x
→
+
(by definition of f + g)
= lim
( )
lim ( )
x
c
x
c
f x
g x
→
+→
(by the theorem on limits)
= f (c) + g(c)
(as f and g are continuous)
= (f + g) (c)
(by definition of f + g)
Hence, f + g is continuous at x = c Proofs for the remaining parts are similar and left as an exercise to the reader Rationalised 2023-24
 MATHEMATICS
114
Remarks
(i)
As a special case of (3) above, if f is a constant function, i e