Chapter 5: Continuity (Sheet 1)
Definitions & Fundamental Properties
1. Definitions of Continuity
| Continuity at a Point $$c$$ | $$\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c)$$ |
| Continuity in Open Interval $$(a, b)$$ | Function is continuous at every point in $$(a, b)$$ |
| Continuity in Closed Interval $$[a, b]$$ | Continuous on $$(a, b)$$AND$$\lim_{x \to a^+} f(x) = f(a)$$AND$$\lim_{x \to b^-} f(x) = f(b)$$ |
2. Algebra of Continuous Functions
Let $$f$$and$$g$$be continuous functions at$$x = c$$.
| Sum Property | $$f + g$$is continuous at$$x = c$$ |
| Difference Property | $$f – g$$is continuous at$$x = c$$ |
| Product Property | $$f \cdot g$$is continuous at$$x = c$$ |
| Scalar Multiplication | $$k \cdot f$$is continuous at$$x = c$$(where$$k$$ is a constant) |
| Quotient Property | $$\frac{f}{g}$$is continuous at$$x = c$$(Provided$$g(c) \neq 0$$) |
| Composition Property | If $$g$$is continuous at$$c$$and$$f$$is continuous at$$g(c)$$, then $$fog$$is continuous at$$c$$ |
3. Differentiability Definitions
| Derivative at a Point $$c$$ | $$f'(c) = \lim_{x \to c} \frac{f(x) – f(c)}{x – c}$$ |
| Left Hand Derivative (LHD) | $$LHD = \lim_{x \to c^-} \frac{f(x) – f(c)}{x – c}$$ |
| Right Hand Derivative (RHD) | $$RHD = \lim_{x \to c^+} \frac{f(x) – f(c)}{x – c}$$ |
| Existence Condition | $$LHD = RHD = \text{Finite Value}$$ |
4. Properties of Differentiability
| Fundamental Theorem | If $$f$$is differentiable at$$c$$$$\Rightarrow$$$$f$$is continuous at$$c$$ |
| Converse Property | If $$f$$is continuous, it is NOT necessarily differentiable (e.g.,$$|x|$$at$$0$$) |