Notes On Algebra of Continuous Functions - CBSE Class 12 Maths

Let f(x) and g(x) be two real functions continuous at real number c.
lim x c f(x) = f(c)     lim x c g(x) = g(c)  .

Now check the continuity of the sum, difference, product and quotient of these functions

f + g, f - g, f . g and f/g

Since f(x) and g(x) are continuous at c, f(x) and g(x) are defined at c.

i) (f + g)(x) is also defined at c.

lim x c (f+g)(x) =  lim x c [f(x) + g(x)]

= lim x c f(x) +  lim x c g(x)

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

lim x c (f+g)(x) = (f+g)(c)

Hence, f + g is continuous at x = c.

ii) (f - g)(x) is also defined at c.

lim x c (f–g)(x) =  lim x c [f(x) – g(x)]

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

= (f - g)(c)

lim x c (f–g)(x) = (f - g)(c)

Hence, f - g is continuous at x = c.

iii) (f . g)(x) is also defined at c.

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

= lim x c f(x) .  lim x c g(x) 

= f(c) . g(c) = (f . g)(c)

lim x c (f.g)(x) =  (f.g)(c)

Hence, f . g is continuous at x = c.

iv) f(x)/g(x), g(c) ≠ 0 is also defined at c. (f/g)(x) = f(x)/g(x)


lim x c (f/g)(x) =  lim x c [f(x) / g(x)]

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

Hence, f/g and g ≠ 0 is continuous at x = c.

Special case:

f(x) = k, where k is a real number.

f(x). g(x) = (k . g)(x) = k . g(x) is also continuous.

lim x c [f(x) . g(x)] lim x c (k.g)(x) = k .  lim x c g(x)

If f(x) is continuous at c and k ∈ R, then k. f(x) is also continuous at c.

k = -1 ⇒ -f is continuous.

If f(x) is continuous at c the, then -f(x) is also continuous at c.

f(x)/g(x) = k/g(x) is also continuous.

If g(x) is continuous at c and k ∈ R, then k/g(x) is also continuous at c provided g(c) ≠ 0.

k = 1 ⇒ 1/g(x) is continuous. The reciprocal of a continuous function is also continuous

Summary

Let f(x) and g(x) be two real functions continuous at real number c.
lim x c f(x) = f(c)     lim x c g(x) = g(c)  .

Now check the continuity of the sum, difference, product and quotient of these functions

f + g, f - g, f . g and f/g

Since f(x) and g(x) are continuous at c, f(x) and g(x) are defined at c.

i) (f + g)(x) is also defined at c.

lim x c (f+g)(x) =  lim x c [f(x) + g(x)]

= lim x c f(x) +  lim x c g(x)

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

lim x c (f+g)(x) = (f+g)(c)

Hence, f + g is continuous at x = c.

ii) (f - g)(x) is also defined at c.

lim x c (f–g)(x) =  lim x c [f(x) – g(x)]

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

= (f - g)(c)

lim x c (f–g)(x) = (f - g)(c)

Hence, f - g is continuous at x = c.

iii) (f . g)(x) is also defined at c.

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

= lim x c f(x) .  lim x c g(x) 

= f(c) . g(c) = (f . g)(c)

lim x c (f.g)(x) =  (f.g)(c)

Hence, f . g is continuous at x = c.

iv) f(x)/g(x), g(c) ≠ 0 is also defined at c. (f/g)(x) = f(x)/g(x)


lim x c (f/g)(x) =  lim x c [f(x) / g(x)]

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

Hence, f/g and g ≠ 0 is continuous at x = c.

Special case:

f(x) = k, where k is a real number.

f(x). g(x) = (k . g)(x) = k . g(x) is also continuous.

lim x c [f(x) . g(x)] lim x c (k.g)(x) = k .  lim x c g(x)

If f(x) is continuous at c and k ∈ R, then k. f(x) is also continuous at c.

k = -1 ⇒ -f is continuous.

If f(x) is continuous at c the, then -f(x) is also continuous at c.

f(x)/g(x) = k/g(x) is also continuous.

If g(x) is continuous at c and k ∈ R, then k/g(x) is also continuous at c provided g(c) ≠ 0.

k = 1 ⇒ 1/g(x) is continuous. The reciprocal of a continuous function is also continuous

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