Continuity, Differentiability and Differentiation

The One Thing

Continuity means that a function approaches its actual value without a break, whereas differentiability means that it has a finite, definite instantaneous slope. Differentiability implies continuity, but continuity alone does not imply differentiability.

Definitions and Results

• Continuity at a Point: A function $f(x)$ is continuous at $x=a$ if $f(a)$ is defined, ${\lim }_{x\to a}f(x)$ exists, and

$$\underset{x\to a}{\lim }f(x)=f(a).$$ The two-sided limit exists only when $$\underset{x\to a^{-}}{\lim }f(x)=\underset{x\to a^{+}}{\lim }f(x).$$

• Left-Hand Limit: The value approached by $f(x)$ as $x$ approaches $a$ from values less than $a$:

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

• Right-Hand Limit: The value approached by $f(x)$ as $x$ approaches $a$ from values greater than $a$:

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

• Continuity on an Interval: A function is continuous on an open interval if it is continuous at every point of that interval. On a closed interval, it must also be right-continuous at the left endpoint and left-continuous at the right endpoint.

• Discontinuity: A point where a function is not continuous. The common types are removable, jump, and infinite discontinuities.

• Differentiability at a Point: A function is differentiable at $x=a$ if its derivative exists there. Equivalently, its left-hand and right-hand derivatives must both exist, be finite, and be equal:

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

• Derivative from First Principles: The derivative of $f(x)$ at $x=a$ is

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

• Geometrical Meaning of Derivative: $f^{\prime }(a)$ is the slope of the tangent to the graph of $y=f(x)$ at $x=a$.

• Physical Meaning of Derivative: A derivative is an instantaneous rate of change. For example, velocity is the derivative of displacement with respect to time.

• Differentiability Implies Continuity: If $f$ is differentiable at $x=a$, then it is necessarily continuous at $x=a$.

• Converse Statement: Continuity does not guarantee differentiability. The function

$$f(x)=|x|$$ is continuous at $x=0$, but it is not differentiable there because its left-hand and right-hand derivatives are unequal.

• Derivative Function: If $f$ is differentiable at every point in an interval, the function assigning $f^{\prime }(x)$ to each $x$ is called the derivative of $f$.

• Standard Derivatives:

$$\frac{d}{dx}(c)=0,\frac{d}{dx}(x^n)=n{x}^{n-1},$$ $$\frac{d}{dx}(\sin x)=\cos x,\frac{d}{dx}(\cos x)=-\sin x,\frac{d}{dx}(e^x)=e^x.$$ Also, $$\frac{d}{dx}(cf(x))=c{f}^{\prime }(x),$$ where $c$ is constant, and $$\frac{d}{dx}[u(x)\pm v(x)]=u^{\prime }(x)\pm v^{\prime }(x).$$

• Product Rule: For $y=uv$,

$$\frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}.$$

• Quotient Rule: For $y=u/v$,

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

• Chain Rule: For $y=f(g(x))$,

$$\frac{dy}{dx}=f^{\prime }(g(x))g^{\prime }(x).$$ The outer function is differentiated first, then multiplied by the derivative of the inner function.

• Implicit Differentiation: When $x$ and $y$ are related by an equation rather than by an explicit expression for $y$, differentiate both sides with respect to $x$ and solve for $dy/dx$.

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

• Differentiation of Inverse Functions: If $y=f^{-1}(x)$, then

$$\frac{dy}{dx}=\frac{1}{f^{\prime }(y)},$$ provided $f$ is invertible and $f^{\prime }(y)\ne 0$.

• Parametric Differentiation: If $x$ and $y$ are functions of a parameter $t$, then

$$\frac{dy}{dx}=\frac{dy/dt}{dx/dt},$$ provided $dx/dt\ne 0$.

• Higher-Order Derivatives: The derivative of $f^{\prime }(x)$ is the second derivative $f^{\prime \prime }(x)$. Repeated differentiation gives higher-order derivatives such as $f^{\prime \prime \prime }(x)$.

• Derivative of an Absolute-Value Function: Absolute-value functions can be continuous but fail to be differentiable at sharp corners, as with $f(x)=|x|$ at $x=0$.

• Further Standard Derivatives:

$$\frac{d}{dx}(\tan x)={\sec }^{2}x,\frac{d}{dx}(\cot x)=-{\mathrm{cosec}}^{2}x,$$ $$\frac{d}{dx}(\sec x)=\sec x\tan x,\frac{d}{dx}(\mathrm{cosec}x)=-\mathrm{cosec}x\cot x.$$ For exponential and logarithmic functions, $$\frac{d}{dx}(e^x)=e^x,\frac{d}{dx}(a^x)=a^x\ln a$$ where $a>0$ and $a\ne 1$, and $$\frac{d}{dx}(\ln x)=\frac{1}{x}(x>0),\frac{d}{dx}({\log }_{a}x)=\frac{1}{x\ln a}.$$ For inverse trigonometric functions, $$\frac{d}{dx}({\sin }^{-1}x)=\frac{1}{\sqrt{1-x^2}},|x|<1,$$ $$\frac{d}{dx}({\cos }^{-1}x)=-\frac{1}{\sqrt{1-x^2}},|x|<1,$$ $$\frac{d}{dx}({\tan }^{-1}x)=\frac{1}{1+x^2},$$ $$\frac{d}{dx}({\cot }^{-1}x)=-\frac{1}{1+x^2},$$ using the standard principal-value convention, and $$\frac{d}{dx}({\sec }^{-1}x)=\frac{1}{|x|\sqrt{x^2-1}},|x|>1,$$ $$\frac{d}{dx}({\mathrm{cosec}}^{-1}x)=-\frac{1}{|x|\sqrt{x^2-1}},|x|>1.$$

• Tangent and Normal: For $y=f(x)$, the tangent at $x=a$ has slope $f^{\prime }(a)$ and equation

$$y-f(a)=f^{\prime }(a)(x-a).$$ If $f^{\prime }(a)\ne 0$, the normal has equation $$y-f(a)=-\frac{1}{f^{\prime }(a)}(x-a).$$

• Increasing and Decreasing Functions: If $f^{\prime }(x)>0$ on an interval, $f$ is increasing there. If $f^{\prime }(x)<0$, $f$ is decreasing there.

• Second Derivative in Applications: The second derivative describes the rate of change of the first derivative. In motion, it represents acceleration when position is differentiated twice with respect to time.

Worked Methods

Testing Continuity at $x=a$

• Verify that $f(a)$ exists.
• Calculate the left-hand limit ${\lim }_{x\to a^{-}}f(x)$.
• Calculate the right-hand limit ${\lim }_{x\to a^{+}}f(x)$.
• Check that the one-sided limits are equal, so that ${\lim }_{x\to a}f(x)$ exists.
• Check that the common limit equals $f(a)$.

For a piecewise function at a joining point, these one-sided checks determine whether the pieces meet continuously.

Testing Differentiability at $x=a$

• Calculate the left-hand derivative:

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

• Calculate the right-hand derivative:

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

• Confirm that both derivatives exist, are finite, and are equal.

For a piecewise function, continuity and differentiability at the joining point must be checked separately: continuity checks whether the function values meet, while differentiability checks whether the slopes also agree.

Differentiating from First Principles

• Start with

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

• Substitute the given function into $f(a+h)$ and $f(a)$.
• Simplify the quotient.
• Evaluate the limit as $h\to 0$, provided the limit exists.

The derivative is obtained as the limiting slope of a secant line and therefore measures instantaneous rather than average change.

Applying Standard Rules

• Identify constants, sums, products, quotients, powers, and elementary functions.
• Apply the relevant rule:

$$\frac{d}{dx}(c)=0,\frac{d}{dx}(x^n)=n{x}^{n-1},$$ together with the constant-multiple and sum-or-difference rules.

• Use the listed trigonometric, exponential, logarithmic, or inverse-function derivatives.
• State any required domain condition, such as $x>0$, $|x|<1$, or $|x|>1$.

Applying the Product and Quotient Rules

For a product $y=uv$:

• Identify $u$ and $v$.
• Differentiate each factor.
• Substitute into

$$\frac{dy}{dx}=u{v}^{\prime }+v{u}^{\prime }.$$

For a quotient $y=u/v$:

• Identify the numerator $u$ and denominator $v$.
• Differentiate both.
• Substitute into

$$\frac{dy}{dx}=\frac{v{u}^{\prime }-u{v}^{\prime }}{v^2}.$$

• Ensure $v\ne 0$.

Applying the Chain Rule

• Identify the inner function $g(x)$.
• Identify the outer function $f$.
• Differentiate the outer function while retaining $g(x)$.
• Multiply by $g^{\prime }(x)$:

$$\frac{d}{dx}[f(g(x))]=f^{\prime }(g(x))g^{\prime }(x).$$

Implicit Differentiation

• Differentiate both sides with respect to $x$.
• Treat $y$ as a function of $x$, so derivatives involving $y$ introduce $dy/dx$.
• Collect all terms containing $dy/dx$.
• Solve for $dy/dx$.

Logarithmic Differentiation

• Take logarithms of both sides of the equation.
• Use logarithmic laws to separate products, quotients, or variable powers.
• Differentiate implicitly.
• Solve for $dy/dx$.
• Substitute the original expression for $y$, if required.

Inverse-Function Differentiation

• Express the relation as $y=f^{-1}(x)$.
• Use

$$\frac{dy}{dx}=\frac{1}{f^{\prime }(y)}.$$

• Confirm that $f$ is invertible and $f^{\prime }(y)\ne 0$.

Parametric Differentiation

• Differentiate $x$ and $y$ separately with respect to $t$.
• Form the quotient

$$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}.$$

• Confirm that $dx/dt\ne 0$.

Finding Tangents, Normals, and Monotonicity

• Calculate $f^{\prime }(a)$.
• Use $f^{\prime }(a)$ as the tangent slope.
• Substitute into

$$y-f(a)=f^{\prime }(a)(x-a)$$ for the tangent.

• If $f^{\prime }(a)\ne 0$, use

$$y-f(a)=-\frac{1}{f^{\prime }(a)}(x-a)$$ for the normal.

• Determine intervals of increase and decrease from the signs of $f^{\prime }(x)$.

Where It Goes Wrong

• Treating continuity as sufficient for differentiability; $f(x)=|x|$ is continuous at $x=0$ but has unequal left-hand and right-hand derivatives there.
• Forgetting that continuity requires all three conditions: $f(a)$ exists, the two-sided limit exists, and the limit equals $f(a)$.
• Assuming a two-sided limit exists without checking that the left-hand and right-hand limits are equal.
• Applying the chain rule without identifying both the outer and inner functions or without multiplying by the derivative of the inner function.
• Using the quotient rule without preserving the order $v{u}^{\prime }-u{v}^{\prime }$, or without requiring $v\ne 0$.
• Omitting domain conditions for inverse trigonometric, logarithmic, exponential, and inverse-function derivatives, or overlooking failure of differentiability caused by a corner, cusp, vertical tangent, or discontinuity.

What Gets Asked

This material supports questions requiring students to:

• Test whether a function is continuous at a point or on an interval.
• Evaluate left-hand and right-hand limits and classify discontinuities as removable, jump, or infinite.
• Determine differentiability using first principles or one-sided derivatives.
• Explain why differentiability implies continuity but continuity does not imply differentiability, using $f(x)=|x|$ at $x=0$.
• Differentiate expressions using standard, constant-multiple, sum, product, quotient, and chain rules.
• Differentiate composite functions after identifying their outer and inner functions.
• Use implicit, logarithmic, inverse-function, or parametric differentiation.
• Find higher-order derivatives, including $f^{\prime \prime }(x)$ and $f^{\prime \prime \prime }(x)$.
• Find equations of tangents and normals at $x=a$.
• Determine intervals on which a function is increasing or decreasing from the sign of $f^{\prime }(x)$.
• Interpret derivatives geometrically as tangent slopes and physically as instantaneous rates of change, including velocity and acceleration.
• Check continuity and differentiability at the joining point of a piecewise function.