Continuity and Differentiability - Continuity and differentiability

Continuity at a Point: A function $f(x)$ is continuous at $x = c$ if the limit of the function as $x$ approaches $c$ exists and is equal to the value of the function at $c$. This requires that the Left Hand Limit (LHL), Right Hand Limit (RHL), and the function value $f(c)$ are all equal: $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c)$. Visually, this means you can draw the graph through that point without lifting your pen; there are no 'holes', 'jumps', or 'vertical asymptotes'.

Continuity in an Interval: A real-valued function $f$ is continuous in an open interval $(a, b)$ if it is continuous at every point in that interval. For a closed interval $[a, b]$, it must also be continuous from the right at $x=a$ and from the left at $x=b$. Geometrically, the graph represents a single, unbroken curve over that specific range of $x$-values.

Differentiability: A function $f(x)$ is differentiable at a point $x = c$ in its domain if the derivative $f'(c)$ exists. This is true if the Left Hand Derivative (LHD) equals the Right Hand Derivative (RHD). Visually, differentiability implies 'smoothness'; the graph has a unique tangent line at that point. If the graph has a sharp corner (like a 'V' shape) or a vertical tangent, it is not differentiable there.

Relation between Continuity and Differentiability: A fundamental theorem states that if a function is differentiable at a point, it must also be continuous at that point. However, the converse is not always true. For example, the absolute value function $f(x) = |x|$ is continuous at $x=0$, but it is not differentiable there because it has a sharp corner, meaning the LHD ($-1$) does not equal the RHD ($1$).

Chain Rule for Composite Functions: If a function $y$ is a composite of two functions $f$ and $g$, such that $y = (f \circ g)(x) = f(g(x))$, its derivative is calculated as the derivative of the outer function with respect to the inner function, multiplied by the derivative of the inner function. Formulaically, $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$, where $u = g(x)$.

Rolle's Theorem: For a function $f$ that is continuous on the closed interval $[a, b]$, differentiable on the open interval $(a, b)$, and satisfies $f(a) = f(b)$, there must exist at least one point $c$ in $(a, b)$ such that $f'(c) = 0$. Visually, if the function starts and ends at the same height, there must be at least one peak or valley where the tangent line is horizontal.

Mean Value Theorem (MVT): If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, there exists a point $c$ in $(a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$. Geometrically, this means there is a point on the curve where the slope of the tangent is equal to the slope of the secant line passing through the endpoints $(a, f(a))$ and $(b, f(b))$.

Logarithmic and Exponential Differentiation: Functions of the form $y = [f(x)]^{g(x)}$ are best solved by taking the natural logarithm ($\\ln$) on both sides to transform the power into a product. This simplifies the differentiation process using the product rule and chain rule.