Continuity and the Intermediate Value Theorem

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Definition of Continuity

Definition: A function $f$ is continuous at a number $x=a$ if $\displaystyle \lim_{x \rightarrow a} f(x) = f(a)$.

Remember that $\displaystyle{\lim_{x \to a} f(x)}$ describes both what is happening when $x$ is slightly less than $a$ and what is happening when $x$ is slightly greater than $a$. Thus there are three conditions inherent in this definition of continuity. A function is continuous at $a$ if the limit as $x\to a$ exists, and $f(a)$ exists, and this limit is equal to $f(a)$. This means that the following three values are equal: $$\lim_{x \to a^-} f(x)\qquad=\qquad f(a)\qquad=\qquad\lim_{x \to a^+} f(x)$$ I.e. the value as $x$ approaches $a$ from the left is the same as the value as $x$ approaches $a$ from the left (the limit exists) which is the same as the value of $f$ at $a$.

If any of these quantities is different, or if any of them fails to exist, then we say that $f(x)$ is discontinuous at $x=a$, or that $f(x)$ has a discontinuity at $x=a$.

What does this mean graphically? If you trace $f$ with a pencil from left to right, as you approach $x=a$, you are at some height $L$ (because $\displaystyle\lim_{x\to a^-}f(x)=L$. As you go through the $x$-value $a$, your height is also $L$ (because $f(a)=L$). Now as you keep going with your pencil, you are beginning this last stretch at height $L$ (because $\displaystyle\lim_{x\to a^+}f(x)=L$).

DO: Sketch $f(x)=\sqrt x$ and let $a=4$. Find $f(a), \displaystyle\lim_{x\to a^-}f(x)=L$, and $\displaystyle\lim_{x\to a^+}f(x)$. Now, follow along your graph as stated in the previous paragraph, looking at each condition as you go. Is $f(x)=\sqrt x$ continuous at $x=a$?

Now, more interestingly, consider $\displaystyle\frac{x^2-1}{x-1}$ in this video:

Definition: A function $f$ is
continuous from the right at $x=a$ if $\displaystyle \lim_{x \rightarrow a^+} f(x) = f(a)$, and is
continuous from the left at $x=a$ if $\displaystyle \lim_{x \rightarrow a^-} f(x) = f(a)$, and is
continuous on an interval $I$ if it is continuous at each interior point of $I$, is continuous from the right at the left endpoint (if $I$ has one), and is continuous from the left at the right endpoint (if $I$ has one).

The Six Pillars of Calculus

Trigonometry Review

Exponential Functions

Logarithms and Inverse functions

Intro to Limits

Limit Laws and Computations

Continuity and the Intermediate Value Theorem

Limits at Infinity

Rates of Change

The Derivative Function

Basic Differentiation Rules

Product and Quotient Rules

Derivatives of Trig Functions

The Chain Rule

Implicit Differentiation

Derivatives of Logs

Derivatives in Science

Related Rates

Linear Approximation and Differentials

Differentiation Review

Absolute and Local Extrema

The Mean Value and other Theorems

$f$ vs. $f'$

Indeterminate Forms and L'Hospital's Rule

Optimization

Newton's Method

Anti-derivatives

The Area under a curve

Definite Integrals

The Fundamental Theorem of Calculus

Definition of Continuity