Algebra of continuous functions is defined for the four arithmetic operations:
- Addition of continuous functions
- Subtraction of continuous functions
- Multiplication of continuous functions
- Division of continuous functions
If two functions are continuous at a point, then the algebraic operations between two functions are also continuous. Let us understand the algebra of continuous functions with the respective theorem and proof. Also, we will solve examples to understand the concept better.
Also, read: Limits And Continuity
Addition of Continuous Functions
Theorem: Let us say, f and g are two real functions that are continuous at a point ‘a’, where ‘a’ is a real number. Then the addition of the two functions f and g is also continuous at ‘a’.
f(x) + g(x) is continuous at x = a |
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Proof: Given,
lim_{x→a} f(x) = f(a)
lim_{x→ a} g(x) = g(a)
Now as per the theorem,
lim_{x → a} (f+g)(x) ⇒ lim_{ x → c} [f(x) + g(x)]
⇒ lim_{x → c} f(x) + lim_{x → c} g(x)
⇒ f(a) + g(a)
⇒ (f + g)(a)
Therefore,
lim_{x → a} (f+g)(x) = (f + g)(c)
Hence, f+g is continuous at x = a.
Subtraction of Continuous Functions
Theorem: Let us say, f and g are two real functions that are continuous at a point ‘a’, where ‘a’ is a real number. Then the subtraction of the two functions f and g is also continuous at ‘a’.
f(x) – g(x) is continuous at x = a |
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Proof: Given,
lim_{x→a} f(x) = f(a)
lim_{x→ a} g(x) = g(a)
Now as per the theorem,
lim_{x → a} (f – g)(x) ⇒ lim_{ x → c} [f(x) – g(x)]
⇒ lim_{x → c} f(x) – lim_{x → c} g(x)
⇒ f(a) – g(a)
⇒ (f – g)(a)
Therefore,
lim_{x → a} (f – g)(x) = (f – g)(c)
Hence, f – g is continuous at x = a.
Multiplication of Continuous Functions
Theorem: If f and g are two real functions that are continuous at a point ‘a’, where ‘a’ is a real number. Then the product of the two functions f and g is also continuous at ‘a’.
f(x) . g(x) is continuous at x = a |
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Proof: Given,
lim_{x→a} f(x) = f(a)
lim_{x→ a} g(x) = g(a)
So, the limit of product of two functions, f and g at x is given by:
lim_{x → a} (f . g)(x) ⇒ lim_{ x → c} [f(x) . g(x)]
⇒ lim_{x → c} f(x) . lim_{x → c} g(x)
⇒ f(a) . g(a)
⇒ (f . g)(a)
Therefore,
lim_{x → a} (f . g)(x) = (f . g)(c)
Hence, f . g is continuous at x = a.
Division of Continuous Function
Theorem: Suppose, f and g are two real functions that are continuous at a point ‘a’, where ‘a’ is a real number. Then the division of the two functions f and g will remain continuous at ‘a’.
f(x) ÷ g(x) is continuous at x = a |
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Proof: Given,
lim_{x→a} f(x) = f(a)
lim_{x→ a} g(x) = g(a)
Now as per the theorem,
lim_{x → a} (f+g)(x) ⇒ lim_{ x → c} [f(x) ÷ g(x)]
⇒ lim_{x → c} f(x) ÷ lim_{x → c} g(x)
⇒ f(a) ÷ g(a)
⇒ (f ÷ g)(a)
Therefore,
lim_{x → a} (f ÷ g)(x) = (f ÷ g)(c)
Hence, f ÷ g is continuous at x = a.
Algebra of Composite Functions
Suppose f and g are real valued functions such that (f o g) is defined at a. If g is continuous at a and if f is continuous at g (a), then (f o g) is continuous at a.
This theorem states that:
f(g(x)) and g(f(x)) are continuous at x = a |
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Solved Example
Q.1: Prove that f(x) = tan x is a continuous function.
Solution: Given,
f(x) = tan x
Since, we know that, by trigonometry formulas,
Tan x = sin x/cos x
Thus,
f(x) = sin x/cos x
Now, this function is defined for all real numbers and cos x ≠ 0, x ≠ (2n+1)π/2.
Sinx and cos x both are continuous functions, therefore, tan x is also continuous, since it is the quotient of sinx/cosx.
Q.2: Prove that the function defined by f(x) = | cos x | is a continuous function.
Solution: Given,
f(x) = |cos x|
f(x) is the real function for all real numbers ‘x’ and the domain of f(x) is the real number
Let g(x) = cos x
And h(x) = |x|
g(x) and h(x) are cosine functions and modulus functions are continuous for all real numbers.
Now,
(goh) (x) = g (h(x)) = g(|x|) = cos |x| is a composite function, hence is a continuous function. But it is not equal to f(x).
Again,
(hog) (x) = h(g(x)) = g(cos x) = |cos x| = f(x) [Given]
Hence,
f(x) = |cos x| = hog (x) is a composition function of two continuous functions. Therefore, it is also a continuous function.
Frequently Asked Questions on Algebra of Continuous Functions
What is the addition and subtraction of two continuous functions?
If two functions are continuous at a point, then the addition and subtraction of two functions is also continuous at the same point, respectively.
What is the multiplication and division of two continuous functions?
Two functions f(x) anf g(x) are continuous at point a, then the product and division of f(x) and g(x) is continuous at the same point.
What is the composition of functions?
If f(x) and g(x) are continuous functions at c, then the composition of two functions (f o g) (x) and (g o f) (x) are continuous functions at c.
Are all rational functions continuous?
All rational functions are continuous.