Continuity Formulas

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1. Continuity of a function at a point
A function f(x) is said to be continuous at a point x = a
i.e. If right hand limit at ‘a’ = left hand limit at ‘a’ = value of the function at ‘a’.
If \(\lim _{x \rightarrow a^{+}}\)f(x) = \(\lim _{x \rightarrow a^{-}}\)f(x) = f(a)

• f(x) is said to be continuous from the left at x = a if \(\lim _{x \rightarrow a^{-}}\) f(x) = f(a).
• f(x) is said to be continuous from the right at x = a if \(\lim _{x \rightarrow a^{+}}\) f(x) = f(a).
• If \(\lim _{x \rightarrow a}\) f(x) does not exist or \(\lim _{x \rightarrow a}\) f(x) ≠ f(a), then f(x) is said to be discontinuous at x = a.

2. Continuity of a function in an interval
(a) A function f(x) is said to be continuous in an open interval (a, b) if it is continuous at every point in (a, b).

(b) A function f(x) is said to be continuous in the closed interval [a, b] if it is

• Continuous at every point of the open interval (a, b).
• Right continuous at x = a.
• Left continuous at x = b.

3. Continuous functions
A function is said to be continuous function if it is continuous at every point in its domain. Following are examples of some continuous function.

• f(x) = x (Identity function)
• f(x) = C (Constant function)
• f(x) = x²
• f(x) = a₀xⁿ + a₀x^n-1+ ….+ aⁿ (Polynomial)
• f(x) = |x|, x + |x|, x – |x|, x|x|
• f(x) = sin x, f(x) = cos x
• f(x) = e^x, f(x) = a^x, a > 0
• f(x) = log x, f(x) = log_ax, a > 0
• f(x) = sinh x, cosh x, tanh x

4. Discontinuous functions

• f(x) = 1/x at x = 0
• f(x) = e^1/x at x = 0
• f(x) = sin (1/x), f(x) = cos(1/x) at x = 0
• f(x) = [x] at every integer
• f(x) = x – [x] at every integer
• f(x) = tan x, f(x) = sec x when x = (2n + 1) \(\frac{\pi}{2}\), n ∈ Z.
• f(x) = cot x, f(x) = cosec x when x = nπ, n ∈ Z.
• f(x) = coth x, f(x) = cosech x at x = 0.

5. Properties of continuous function
If f(x) and g(x) are continuous functions then following are also continuous functions:

• f(x) + g(x)
• f(x) – g(x)
• f(x). g(x)
• λf(x), where λ is a constant
• f(x)/g(x), if g(x) ≠ 0
• f[g(x)]

6. Some Important points
(i) When we say that the function f(x) is continuous at a point x = a, it mean that at point (a, f(a)) graph is untraken.

(ii) Kinds of discontinuity

• \(\lim _{x \rightarrow a^{-}}\) f(x) = \(\lim _{x \rightarrow a^{+}}\) f(x), then f is said to have non “removal discontinuity” of first kind.
• \(\lim _{x \rightarrow a^{-}}\) f(x) ≠ \(\lim _{x \rightarrow a^{+}}\) f(x), then f is said to have non removal discontinuity of first kind.
• At least one of \(\lim _{x \rightarrow a^{-}}\) f(x) or \(\lim _{x \rightarrow a^{+}}\) f(x) does not exist then f is said to have discontinuity of 2^nd kind at x = a
• Continuity of composite function: If the function u = f(x) is continuous at the point x = a and the function y = g(u) is continuous at the point u = f(a) then composite function y = (gof)(x) = g(f(x)) is continuous at point x = a

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