Definitions [5]
A function f(x) is said to be discontinuous at x = a if it is not continuous at x = a, i.e.
- \[\lim_{x\to a}f\left(a\right)\] does not exist.
- The left-hand limit and the right-hand limit are not equal.
- \[\lim_{x\to a}f\left(x\right)\neq f\left(a\right)\].
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.
A function f(x) is said to be continuous in its domain if it is continuous at every point in its domain.
- A function f(x) is said to be continuous in an open interval (a, b) if it is continuous for every value of x in the interval (a, b).
- A function f(x) is said to be continuous in the closed interval [a, b] if
(a) It is continuous for every value of x in the open interval (a, b).
(b) f(x) is continuous at x = a from right
i.e.\[\lim_{x\to a^{+}}f\left(x\right)=f\left(a\right)\]
(c) f(x) is continuous at x = b from left,
i.e. \[\lim_{x\to a^{-}}f\left(x\right)=f\left(b\right)\].
Theorems and Laws [2]
Intermediate value theorem for continuous function If f is a continuous function on a closed interval [a, b] and if y₀ is any value between f(a) and f(b), then y₀ = f(c) for some c in [a, b].
There are some functions which are always continuous in their respective domain.
• Every constant function is continuous.
• Every identity function is continuous.
• Every rational function is continuous.
• Modulus function f(x) = ∣x∣ is continuous.
If g is continuous at c, and f is continuous at g(c), then their composite function \[(f \circ g)\], defined as \[(f \circ g)(x) = f(g(x))\], is also continuous at c.
Key Points
- 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.
| Type of Discontinuity | Exact Point (x = c) | Definition |
|---|---|---|
| Removable Discontinuity | \[\lim_{x\to c}f(x)\] exists, but f(c) is either not defined or not equal to the limit | A hole in the function, where the limit exists but does not match the function value. |
| Jump Discontinuity | \[\lim_{x\to c^{-}}f(x)\neq\lim_{x\to c^{+}}f(x)\] | The function has a sudden jump in value |
| Infinite (Essential) Discontinuity | \[\lim_{x\to c}f(x)=\pm\infty\] | The function approaches a vertical asymptote. |
| Oscillatory Discontinuity | The function fluctuates indefinitely as \[x\rightarrow c\] | The function oscillates near the point |
- If \[f\] and \[g\] are continuous at \[c\], then \[f+g\], \[f-g\], and \[fg\] are continuous at \[c\].
- \[\dfrac{f}{g}\] is continuous provided
\[ g(c) \neq 0. \] - If \[f\] is continuous, then \[\lambda f\] is continuous.
- If \[g\] is continuous and non-zero, then \[ \frac{1}{g} \] is continuous.
- Rational functions are continuous on their domains.
- sin x and cos x are continuous for all real \[x\].
- \[\tan x\] is continuous wherever \[\cos x \neq 0\].
- If \[g\] is continuous at \[c\] and \[f\] is continuous at \[g(c)\], then \[ f \circ g \] is continuous at \[c\].
