Continuity, Differentiability, and Fundamental Calculus Theorems

Continuity at a Point

• Precise Three-Part Definition of Continuity:

  • A function $f$ is continuous at a specific point $a$ (the point of interest) if and only if all three of the following conditions are simultaneously satisfied:

  • 1. $f(a)$ is defined:

    • The value $a$ must reside within the domain of the function $f$.

    • If $a$ is not in the domain of $f$, the function undefined at that coordinate cannot be continuous there.

  • 2. $\lim_{x \to a} f(x)$ exists:

    • The two-sided limit requires that the function approaches the same finite real number from both the left and the right sides:         $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$

    • If the left-hand limit and right-hand limit disagree, or if the function grows without bound toward positive or negative infinity, the limit fails to exist.

  • 3. $\lim_{x \to a} f(x) = f(a)$

    • The limiting value approached by the function as $x$ approaches $a$ must exactly equal the actual value obtained by evaluating the function at $a$.

• Analytical Consequences of Violating Continuity Conditions:

  • If condition 1 fails, the graph displays a missing point (a hole) or an asymptote.

  • If condition 2 fails, the graph exhibits a jump discontinuity or an infinite/oscillatory discontinuity.

  • If condition 3 fails while conditions 1 and 2 hold, the graph exhibits a removable discontinuity where a single point is displaced from the curve.

Differentiability

• Necessary Conditions for Differentiability at a Point:

  • For a function $f$ to be differentiable at a point $a$ (the point of interest), it must satisfy four core conditions:

  • 1. Continuous at $a$ (the point of interest):

    • Continuity is a strict prerequisite for differentiability. A function cannot possess a derivative at any point where it fails to be continuous.

  • 2. $\lim_{x \to a} f'(x)$ exists (and both left and right sides are equal):

    • The derivative limit must exist from both directions, meaning the left-hand derivative and right-hand derivative must be finite and equal:         $\lim_{x \to a^-} \frac{f(x) - f(a)}{x - a} = \lim_{x \to a^+} \frac{f(x) - f(a)}{x - a}$

  • 3. No sharp corners or cusps:

    • A sharp corner occurs where the left-hand and right-hand slopes approach different finite values.

    • A cusp occurs where slopes approach opposite infinities.

    • The presence of either feature causes the limit defining the derivative to fail to exist.

  • 4. No vertical tangents:

    • A vertical tangent occurs when the slope of the tangent line approaches infinity ($\infty$ or $-\infty$).

    • Because infinite slope is not a real number, the derivative does not exist at points featuring a vertical tangent line.

Domain and Differentiability of Particular Functions

• Differentiability Domain of $\frac{1}{x}$ and $|x|$:

  • The functions $\frac{1}{x}$ and $|x|$ are differentiable on their domain, which is:     $(-\infty, 0) \cup (0, \infty)$

• In-Depth Analysis of $f(x) = \frac{1}{x}$:

  • Domain:     $(-\infty, 0) \cup (0, \infty)$

  • The value $x = 0$ is excluded from the domain because division by zero is undefined.

  • At every point where $x \neq 0$, the derivative is given by:     $f'(x) = -\frac{1}{x^2}$

  • Because $f(x)$ is undefined at $x = 0$, it is discontinuous and non-differentiable at $0$, but differentiable everywhere else in its domain.

• In-Depth Analysis of $f(x) = |x|$:

  • The absolute value function is defined for all real numbers, but its differentiability domain is:     $(-\infty, 0) \cup (0, \infty)$

  • Evaluation at the corner point $x = 0$:

  • For $x > 0$, $f(x) = x$ and the derivative is $1$.

  • For $x < 0$, $f(x) = -x$ and the derivative is $-1$.

  • Because the left-hand limit ($-1$) and right-hand limit ($1$) do not match, a sharp corner exists at $x = 0$.

  • Consequently, $|x|$ is continuous everywhere on $(-\infty, \infty)$, but differentiable only on $(-\infty, 0) \cup (0, \infty)$.

Intermediate Value Theorem (IVT)

• Statement and Hypotheses:

  • Let $f$ be a function that is continuous (cts) on the closed interval $[a, b]$.

  • Let $N$ be any number between $f(a)$ and $f(b)$, where:     $f(a) \neq f(b)$

• Conclusion:

  • Then there exists a number $c$ in the open interval $(a, b)$ such that:     $f(c) = N$

• Geometric and Conceptual Meaning:

  • A continuous function cannot transition from one output value to another without passing through every single intermediate value between them.

  • On the coordinate plane, any horizontal line $y = N$ situated between $y = f(a)$ and $y = f(b)$ must cross the graph of $y = f(x)$ at least once on the interval $(a, b)$.

  • This theorem forms the mathematical foundation for root-finding: if $f(a)$ and $f(b)$ have opposite signs, there is guaranteed to be at least one root $c \in (a, b)$ where $f(c) = 0$.

Extreme Value Theorem (EVT)

• Statement and Hypotheses:

  • If a function ($f^n$) $f$ is continuous (cts) on a closed interval $[a, b]$:

• Conclusion:

  • Then $f$ attains an absolute maximum $f(c)$ and an absolute minimum $f(d)$ at some numbers $c$ and $d$ in $[a, b]$.

• Analytical Requirements and Significance:

  • Both conditions—continuity and a closed interval—are mandatory:

  • If the interval is open, such as $(a, b)$, the function may approach an upper or lower supremum without ever achieving it as an attained maximum or minimum value.

  • If the function contains a discontinuity inside $[a, b]$, the extreme values can fail to exist entirely.

  • The locations $c$ and $d$ where absolute extremes occur will always be found either at the endpoints of the interval ($x = a$ or $x = b$) or at critical numbers in the interior open interval $(a, b)$ where $f'(x) = 0$ or $f'(x)$ is undefined.

Rolle's Theorem

• Statement and Hypotheses:

  • Let $f$ be a function ($f^n$) that satisfies the following three hypotheses:

  • 1. $f$ is continuous (cts) on the closed interval $[a, b]$

  • 2. $f$ is differentiable on the open interval $(a, b)$

  • 3. $f(a) = f(b)$

• Conclusion:

  • Then there is a number $c$ in the open interval $(a, b)$ such that:     $f'(c) = 0$

• Geometric Interpretation:

  • If a smooth, continuous path begins and ends at identical heights on a graph, the curve must either be completely horizontal throughout, or turn around at least once to return to its initial elevation.

  • At that turnaround point $c$, the tangent line to the curve is completely horizontal, with a slope of zero.

Mean Value Theorem (MVT)

• Statement and Hypotheses:

  • Let $f$ be a function ($f^n$) that satisfies the hypotheses:

  • 1. $f$ is continuous (cts) on the closed interval $[a, b]$

  • 2. $f$ is differentiable on the open interval $(a, b)$

• Conclusion:

  • Then there is a number $c$ in the open interval $(a, b)$ such that:     $f'(c) = \frac{f(b) - f(a)}{b - a}$

• Geometric and Physical Meaning:

  • The term on the right represents the average rate of change over the interval, or the slope of the secant line connecting $(a, f(a))$ and $(b, f(b))$:     $\text{Secant Slope} = \frac{f(b) - f(a)}{b - a}$

  • The term on the left represents the instantaneous rate of change at the point $c$, or the slope of the tangent line:     $\text{Tangent Slope} = f'(c)$

  • The Mean Value Theorem guarantees that there is at least one point $c \in (a, b)$ where the instantaneous rate of change exactly equals the average rate of change, making the tangent line parallel to the secant line.

• Connection to Rolle's Theorem:

  • Rolle's Theorem is the specific case of the Mean Value Theorem where $f(a) = f(b)$.

  • When $f(a) = f(b)$, the average rate of change evaluates to:     $\frac{f(b) - f(a)}{b - a} = \frac{0}{b - a} = 0$

  • Substituting this into the Mean Value Theorem formula yields:     $f'(c) = 0$