12.Continuity

Dated: 30-10-2024

Continuity

There are 3 conditions for a function¹ to be continous near \(x= c\).

\(f(c)\) is defined.
\[\lim_{x \rightarrow c^+} f(x) = \lim_{x \rightarrow c^-} f(x)\]
\[\lim_{x \rightarrow c} f(x) = f(c)\]

Continuous Function

A function¹ is called continuous if it follows the rules of continuity over the interval \((-\infty, +\infty)\).

Discontinuity

If a function¹ violates any of the rules of continuity at point \(c\) then it is called point of discontinuity.

Properties of Continuous Functions

if \(f(x)\) and \(g(x)\) are continuous functions then following functions are also continuous.

\(f(x) + g(x)\)
\(f(x) - g(x)\)
\(f(x) \cdot g(x)\)
\(\frac{f(x)}{g(x)}\)

We can prove these properties by looking at the theorems of limits²

Continuity of Composite Functions

If \(\lim g(x) = L\) and \(f(x)\) is continuous at \(x = L\) then \(\lim{fog(x)} = f(L)\)
If \(g(x)\) is continuous at \(c\) and \(f(x)\) is continuous at \(g(c)\) then \(fog(x)\) is continuous at \(c\).

Theorems

Intermediate Value Theorem

If \(f(x)\) is continuous at the interval \([a, b]\) and there is a value \(c\) between \(f(a)\) and \(f(b)\), inclusively, then there must be a value \(x\) where \(f(x) = c\).

Another Theorem

If \(f(x)\) is continuous over the interval \([a, b]\) and \(f(a)\) and \(f(b)\) have opposite signs then there is at least one solution for \(f(x) = 0\) in the interval \((a, b)\).