Continuity and Differentiability

The One Thing

Continuity concerns whether a function approaches its actual value without a break, jump, or missing point, whereas differentiability concerns its instantaneous rate of change and tangent slope. Differentiability implies continuity, but continuity alone does not imply differentiability; the chain rule and implicit differentiation provide systematic methods for differentiating composite and implicitly defined functions.

Definitions and Results

• Continuity at a Point: A function $f$ is continuous at $x=a$ if

$$\underset{x\to a}{\lim }f(x)=f(a).$$ This requires $f(a)$ to be defined, ${\lim }_{x\to a}f(x)$ to exist, and the limit to equal $f(a)$. Equivalently, $$\underset{x\to a^{-}}{\lim }f(x)=\underset{x\to a^{+}}{\lim }f(x)=f(a).$$

• Continuity on an Interval: A function is continuous on an interval if it is continuous at every point in that interval. One-sided continuity is considered at endpoints.

• Left-Hand Continuity: $f$ is left-continuous at $x=a$ if

$$\underset{x\to a^{-}}{\lim }f(x)=f(a).$$

• Right-Hand Continuity: $f$ is right-continuous at $x=a$ if

$$\underset{x\to a^{+}}{\lim }f(x)=f(a).$$

• Discontinuity: A function is discontinuous at a point if its limit does not exist, its limit differs from its function value, or the function is not defined there.

• Differentiability: $f$ is differentiable at $x=a$ if

$$f^{\prime }(a)=\underset{h\to 0}{\lim }\frac{f(a+h)-f(a)}{h}$$ exists.

• Derivative: $f^{\prime }(x)$ represents the instantaneous rate of change of $f$ with respect to $x$ and gives the slope of the tangent to the curve. From first principles,

$$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h},$$ when the limit exists.

• Left-Hand Derivative:

$$f_{-}^{\prime }(a)=\underset{h\to 0^{-}}{\lim }\frac{f(a+h)-f(a)}{h}.$$

• Right-Hand Derivative:

$$f_{+}^{\prime }(a)=\underset{h\to 0^{+}}{\lim }\frac{f(a+h)-f(a)}{h}.$$

• Differentiability and Continuity: If a function is differentiable at a point, it must be continuous there. The converse is not always true. For example,

$$f(x)=|x|$$ is continuous at $x=0$ but not differentiable there.

• Differentiability Criterion: A function is differentiable at $x=a$ if and only if its left-hand and right-hand derivatives exist and are equal. Differentiability may fail at a corner, cusp, vertical tangent, or discontinuity.

• Composite Function: A composite function is formed by applying one function to the output of another:

$$(f\circ g)(x)=f(g(x)).$$

• Chain Rule: If $y=f(g(x))$, then

$$\frac{dy}{dx}=f^{\prime }(g(x))g^{\prime }(x).$$ Equivalently, if $y$ is a function of $u$ and $u$ is a function of $x$, $$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}.$$

• Implicit Function: An implicit function is a relation connecting $x$ and $y$ by an equation rather than expressing $y$ explicitly as a function of $x$.

• Implicit Differentiation: Differentiate both sides of an equation with respect to $x$, treating $y$ as a function of $x$. Every differentiated term involving $y$ must include $\frac{dy}{dx}$.

• Derivative of Inverse Functions: If $y=f(x)$ has an inverse and $f^{\prime }(x)\ne 0$, then

$$\frac{d}{dx}(f^{-1}(x))=\frac{1}{f^{\prime }(f^{-1}(x))}.$$

• Exponential and Logarithmic Derivatives:

$$\frac{d(e^x)}{dx}=e^x,\frac{d(a^x)}{dx}=a^x\ln (a),$$ $$\frac{d(\ln x)}{dx}=\frac{1}{x},\frac{d({\log }_{a}x)}{dx}=\frac{1}{x\ln a}.$$

• Logarithmic Differentiation: This method differentiates products, quotients, or variable powers by taking logarithms before differentiating.

• Continuity of Common Functions: A polynomial is continuous and differentiable for every real number. Rational functions are continuous wherever their denominators are nonzero.

• Algebra of Continuous Functions: If $f$ and $g$ are continuous at $a$, then $f+g$, $f-g$, and $fg$ are continuous at $a$. The quotient $f/g$ is continuous wherever $g(a)\ne 0$.

• Continuity of Composite Functions: If $g$ is continuous at $a$ and $f$ is continuous at $g(a)$, then $f\circ g$ is continuous at $a$.

• Basic Derivative Rules:

$$\frac{d(c)}{dx}=0,\frac{d(x^n)}{dx}=n{x}^{n-1},$$ $$\frac{d(cf)}{dx}=c{f}^{\prime },\frac{d(f\pm g)}{dx}=f^{\prime }\pm g^{\prime }.$$

• Product Rule:

$$\frac{d(uv)}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}.$$

• Quotient Rule:

$$\frac{d(u/v)}{dx}=\frac{v(du/dx)-u(dv/dx)}{v^2},v\ne 0.$$

• Implicit Differentiation Rules:

$$\frac{d(y^n)}{dx}=n{y}^{n-1}\frac{dy}{dx},$$ $$\frac{d(\sin y)}{dx}=\cos y\frac{dy}{dx}.$$

• Standard Trigonometric Derivatives:

$$\frac{d(\sin x)}{dx}=\cos x,\frac{d(\cos x)}{dx}=-\sin x,$$ $$\frac{d(\tan x)}{dx}={\sec }^{2}x,\frac{d(\cot x)}{dx}=-{\mathrm{cosec}}^{2}x.$$ Also, $$\frac{d(\sec x)}{dx}=\sec x\tan x,\frac{d(\mathrm{cosec}x)}{dx}=-\mathrm{cosec}x\cot x.$$

• Inverse Trigonometric Derivatives: Within their appropriate domains,

$$\frac{d({\sin }^{-1}x)}{dx}=\frac{1}{\sqrt{1-x^2}},$$ $$\frac{d({\cos }^{-1}x)}{dx}=-\frac{1}{\sqrt{1-x^2}},$$ $$\frac{d({\tan }^{-1}x)}{dx}=\frac{1}{1+x^2}.$$

• Differentiability Implies Continuity: The relation

$$f(x)-f(a)=\left[\frac{f(x)-f(a)}{x-a}\right](x-a)$$ shows that, when the derivative limit exists, $f(x)\to f(a)$ as $x\to a$. Therefore, differentiability implies continuity.

Worked Methods

Testing Continuity at a Point

• Check that $f(a)$ is defined.
• Determine whether ${\lim }_{x\to a}f(x)$ exists.
• Compare the limit with the function value.
• Alternatively, calculate the one-sided limits and verify

$$\underset{x\to a^{-}}{\lim }f(x)=\underset{x\to a^{+}}{\lim }f(x)=f(a).$$

Testing Continuity of a Piecewise Function

• Identify the joining point.
• Find the left-hand limit at the joining point.
• Find the right-hand limit.
• Evaluate the function at the joining point.
• Equate the left-hand limit, right-hand limit, and function value.

For continuity at $x=a$, $$\underset{x\to a^{-}}{\lim }f(x)=\underset{x\to a^{+}}{\lim }f(x)=f(a).$$

Testing Differentiability at a Point

• Establish continuity at the point first.
• Calculate the left-hand derivative:

$$f_{-}^{\prime }(a)=\underset{h\to 0^{-}}{\lim }\frac{f(a+h)-f(a)}{h}.$$

• Calculate the right-hand derivative:

$$f_{+}^{\prime }(a)=\underset{h\to 0^{+}}{\lim }\frac{f(a+h)-f(a)}{h}.$$

• Conclude differentiability only if both derivatives exist and are equal.

For a piecewise function, differentiability at the joining point requires continuity there before the left-hand and right-hand derivatives are compared.

Differentiating a Composite Function by the Chain Rule

• Identify the inner function $u=g(x)$.
• Differentiate the outer function with respect to $u$.
• Differentiate $u$ with respect to $x$.
• Multiply:

$$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}.$$

For $y=f(g(x))$, $$\frac{dy}{dx}=f^{\prime }(g(x))g^{\prime }(x).$$

Differentiating an Implicit Function

• Differentiate both sides with respect to $x$.
• Treat $y$ as a function of $x$.
• Apply the chain rule to every occurrence of $y$.
• Collect the terms containing $\frac{dy}{dx}$.
• Solve algebraically for $\frac{dy}{dx}$.

For example, the required rules include $$\frac{d(y^n)}{dx}=n{y}^{n-1}\frac{dy}{dx}$$ and $$\frac{d(\sin y)}{dx}=\cos y\frac{dy}{dx}.$$

Differentiating by Logarithmic Differentiation

• Take logarithms of both sides of the equation.
• Use logarithm laws to separate products, quotients, and powers.
• Differentiate implicitly with respect to $x$.
• Solve for $\frac{dy}{dx}$.
• Substitute the original expression for $y$, if required.

This method is particularly useful for products, quotients, or variable powers.

Establishing Continuity from Known Results

• Identify whether the function is a polynomial, rational function, sum, difference, product, quotient, or composite.
• Apply the relevant continuity result.
• For rational functions, check that the denominator is nonzero.
• For quotients $f/g$, check that $g(a)\ne 0$.
• For composites $f\circ g$, check continuity of $g$ at $a$ and of $f$ at $g(a)$.

Where It Goes Wrong

• Forgetting that continuity requires all three conditions: $f(a)$ must be defined, the limit must exist, and the limit must equal $f(a)$.
• Checking only one-sided limits instead of verifying

$$\underset{x\to a^{-}}{\lim }f(x)=\underset{x\to a^{+}}{\lim }f(x)=f(a).$$

• Testing differentiability without first establishing continuity for a piecewise function.
• Treating continuity as sufficient for differentiability; $f(x)=|x|$ is continuous at $x=0$ but not differentiable there.
• Omitting $\frac{dy}{dx}$ when differentiating terms such as $y^n$ or $\sin y$ during implicit differentiation.
• Applying the quotient rule without retaining the condition $v\ne 0$, or applying continuity of a rational function where its denominator is zero.

What Gets Asked

• Verify whether a function is continuous at $x=a$ using the function value and two-sided or one-sided limits.
• Determine whether a function is continuous on an interval, including one-sided continuity at endpoints.
• Identify and classify discontinuities from the limit, function value, or domain.
• Test continuity and differentiability of a piecewise function at its joining point.
• Use first principles to evaluate

$$f^{\prime }(a)=\underset{h\to 0}{\lim }\frac{f(a+h)-f(a)}{h}.$$

• Compare left-hand and right-hand derivatives to determine differentiability.
• Explain why differentiability implies continuity and why the converse fails, using $f(x)=|x|$.
• Apply basic derivative, product, quotient, and chain rules.
• Differentiate composite functions using

$$\frac{dy}{dx}=f^{\prime }(g(x))g^{\prime }(x).$$

• Perform implicit differentiation, including expressions involving $y^n$ and $\sin y$.
• Find derivatives of inverse functions, exponential and logarithmic functions, trigonometric functions, and inverse trigonometric functions.
• Use logarithmic differentiation for products, quotients, and variable powers.