Preface How is this course different from Calculus?
A large portion of this course will consist in studying real numbers, functions of a real variable, continuity, limits, derivatives and integrals. So what is the difference with the Calculus course? An example will be useful to explain the difference. Suppose Jasmine has just taken the Calculus course, and she is helping Kevin with this problem: If \(f(x)=x^{10}\text{,}\) find \(f'(x)\text{.}\) Jasmine solves the problem and gets the right answer: using the power rule, she finds that \(f'(x)=10x^9\text{.}\) This is what she learned in the Calculus course.
But suppose now Kevin asks: "But what does \(f'(x)\) stand for?". Jasmine will probably answer: "It’s the derivative of \(f(x)\)". This is correct, but it really doesn’t explain what it is, it just says what its name is. If Kevin is curious he will want to know what the derivative is all about and why we are so interested in it. Depending on how well Jasmine learned the Calculus material, she may be able to explain that the derivative is the limit as \(h\) goes to zero of the difference quotient:
\begin{equation}
f'(x)=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h},\tag{0.0.1}
\end{equation}
and it represents the slope of the tangent line to the graph of \(f\text{.}\) But if Kevin were to press her with the next natural question: "but what is a limit? And what is a tangent line?", she would not be able to give a precise answer. And (depending on the type of Calculus course she took) she may not be able to explain why the derivative of \(x^n\) is \(nx^{n-1}\text{.}\) Because in the Calculus course much of the focus is on how to use some rules, without worrying where they come from.
In Real Analysis, we will give a very precise meaning to all notions we meet. So it is not enough to say that \(\displaystyle \lim_{x\rightarrow c}f(x)\) is a number that \(f(x)\) gets closer and closer to when \(x\) gets closer and closer to \(c\text{.}\) Instead, we will give a precise definition that uses only notions to which we can already give a precise meaning. In the example of the limit, the notion of "closer and closer" commonly used in the Calculus course is not clear at all. We may have an intuitive idea, but what precisely do we mean by "closer and closer"?
The other important difference with the Calculus course is that it is not enough to say that the derivative of \(x^n\)is \(nx^{n-1}\) because that’s what the power rule says. We want to know why the power rule is true. Since we have defined the derivative to be the limit (0.0.1), to know that the power rule is true we need to prove that
\begin{equation}
\lim_{h\rightarrow 0}\frac{(x+h)^n-x^n}{h}=nx^{n-1}.\tag{0.0.2}
\end{equation}
In other words, we want to have a proof of all statements we make.
So most of the difference between the Calculus course and the Real Analysis course is that we will give very precise definitions, and provide proofs for the statements made.
This brings us to an important discussion. You will find a number of courses that are listed as prerequisites for this course, such as the Calculus sequence and the Discrete math course. While it will of course be helpful to be already familiar with limits, derivatives and integrals (that you studied in Calculus), in reality we will study all those notions again from scratch and more in depth, without assuming any prior knowledge. But we will need to formulate very precise definitions and provide proofs of the statements. To this end, the Discrete course is a more essential prerequisite. You have been introduced to proofs in that course. In this course we will see how to apply the skills and techniques for writing proofs to gain a real understanding of the various notions contained in the Calculus course, and just as importantly, to understand why all the rules and the theorems are true.
