Revision: Continuity Maths HSC Commerce (English Medium) 11th Standard Maharashtra State Board

A real-valued function \[f\] is said to be continuous at \[x = c\] if

\[ \boxed{\lim_{x \to c} f(x) = f(c)} \]

In terms of one-sided limits,

\[ \boxed{\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c)} \]

Thus, for continuity at \[x = c\]:

• \[f(c)\] must be defined.
• Left-hand limit must exist.
• Right-hand limit must exist.
• Both limits must be equal to \[f(c)\]

If any of these conditions fails, \[f\] is discontinuous at \[x = c\].

A real function \[f\] is said to be a continuous function if it is continuous at every point in its domain.

\[ \boxed{\lim_{x \to c} f(x) = f(c)} \] for every \[c\] in the domain of \[f\].

Continuity at End Points

If \[f\] is defined on a closed interval \[[a, b]\]:

At the left endpoint \[a\], \[ \boxed{\lim_{x \to a^+} f(x) = f(a)} \]

At the right endpoint \[b\], \[ \boxed{\lim_{x \to b^-} f(x) = f(b)} \]

Only the appropriate one-sided limit is considered at an endpoint.

Removable Discontinuity:

If \[\lim_{x\to a^{-}}f\left(x\right)=\lim_{x\to a^{+}}f\left(x\right)\neq f\left(a\right),\] then f(x) is said to be removable discontinuous.

Non Removable Discontinuity:

If \[\lim_{x\to a^{+}}f\left(x\right)\neq\lim_{x\to a^{-}}f\left(x\right),\] then f(x) is said to be non-removable discontinuous.

•  Continuity at \[x = c\]: \[ \boxed{\lim_{x \to c} f(x) = f(c)} \]
• Practical test: \[ \boxed{\text{LHL} = \text{RHL} = f(c)} \]
  If this condition fails, the function is discontinuous at \[c\].
• A function is continuous if it is continuous at every point in its domain.
• Constant, identity and polynomial functions are continuous on their domains.
• \[\dfrac{1}{x}\] is continuous for \[x \neq 0\].
• For a piecewise function, check continuity particularly at the point where the rule changes.
• The greatest integer function \[[x]\] is discontinuous at every integer.