Composition of Continuous and Discontinuous Functions

Continuous Functions and Discontinuous Functions

Continuous Functions: A function f(x) is said to be continuous, if it is continuous at each point of its domain. A function f(x) is said to be everywhere continuous if it is continuous on the entire real line (-∞, ∞) i.e., on R. Some Fundamental Results on Continuous Functions: Here, we list some fundamentals result on continuous functions without giving their proofs.

Result:  Let f(x) and g(x) be two continuous functions on their common domain D and let c be real number. Then

• C f is continuous
• f+ g is continuous
• f – g is continuous
• fg is continuous
• f/g is continuous
• 1/g is continuous
• fⁿ, for all n ϵ N is continuous.

Listed below are some common type of functions that are continuous in their domains.

• Constant Function: Every constant function is every – where continuous.
  
• Identity Function: The identity function I(x) is defined by I(x) = x for all x ϵ R.
  
• Modulus Function: The modulus function f(x) is defined as\(f\left( x \right)=\left| x \right|=\left\{ \begin{align} & x,\,\,\,\,\,if\,\,x\ge 0 \\ & -x,\,\,if\,\,x<0 \\\end{align} \right.\). Clearly, the domain of f(x) is R and this function is everywhere continuous.
  
• Exponential Function: if a is positive real number, other than unity, then the function f(x) defined by f(x) =a^x for all x ϵ R is called the exponential function. The domain of this function id R. it is evident form its graph that it everywhere continuous.
  
• Logarithm Function: If a is positive real number other than unity, then a function by f(x) = logₐ x is called the logarithm function. Clearly its domain is the set of all positive real numbers and it is continuous on its domain.
  
• Polynomial function: A function of the form  f(x) = a₀ + a₁ x +a₂ x² +…..+ a_nx ⁿ where a₀, a₁, a₂, …, a_n is called a polynomial function. This function is everywhere continuous.
• Rational Function: if p(x) and q(x) are two polynomials, then a function, f(x) of the form \(f\left( x \right)=\frac{p\left( x \right)}{q\left( x \right)}\), q(x) ≠ 0 is called a polynomial function.
• This function is continuous on its domain i.e., it is everywhere continuous except at points where q(x) = 0.
• Trigonometric Functions: all trigonometrical functions viz. sinx, cosx, secx, tanx, cotx, cosecx are continuous at each point of their respective domains.

Result: The composition of two continuous functions is a continuous function i.e., f and g are two functions such that g is continuous at a point a and f is continuous at g (a), then fog continuous at a.

Discontinuous Functions: A function f is said to be discontinuous at a point a of its domain D if it is not continuous threat. The point a is then called a point of discontinuity of the function. The discontinuity may arise due to any of the following situations:

• \(\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f\left( x \right)\) or \(\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f\left( x \right)\) of both may not exist.
• \(\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f\left( x \right)\) as well as \(\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f\left( x \right)\) may exist, but are unequal.
• \(\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f\left( x \right)\) as well as \(\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f\left( x \right)\) both exist, but either of the two or both may not be equal to f (a).

We classify the points of discontinuity according to various situations discussed above.

• Removable discontinuity: A function f is said to have removable discontinuity at x = a if \(\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f\left( x \right)=\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f\left( x \right)\) but their common value is not equal to f(a). Such a discontinuity can be removed by assigning a suitable value to the function f at x = a.
• Discontinuity of the first kind: A function f is said to have a discontinuity of the first kind at x = a if \(\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f\left( x \right)\) and \(\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f\left( x \right)\) both exist but are not equal. f is said to have a discontinuity of the first kind from the left at x = a if \(\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f\left( x \right)\) exists but not equal to f(a). Discontinuity of the first kind from the right is similarly defined.
• Discontinuity of second kind: A function f is said to have a discontinuity of the second kind at x = a if neither \(\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f\left( x \right)\) nor \(\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f\left( x \right)\) exists.

A function f is said to have discontinuity of the second kind from the left at x = a if \(\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f\left( x \right)\) does not exist.

Similarly, if \(\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f\left( x \right)\) does not exist, then f is said to have discontinuity of the second kind from the right at x = a.

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