Definitions [5]
A function f(x) is said to be continuous at a point x = a, if the following three conditions are satisfied
- f is defined at every point on an open interval containing a.
- \[\lim_{x\to a}f\left(x\right)\] exists.
- \[\lim_{x\to a}f\left(x\right)=f\left(a\right)\].
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.
- \[\lim_{x\to a}f\left(x\right)\neq f\left(a\right)\].
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.
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.
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
| 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 |
| Operation on continuous functions | Result |
|---|---|
| f + g | Continuous |
| f − g | Continuous |
| fg | Continuous |
| \[\dfrac{f}{g}\] | Continuous |
| \[f \circ g\] | Continuous |
