Continuity and Differentiability

Now we're going to look at two types of smoothness:

• continuity, which means the graph now has to be drawn in one piece, whithout taking the pen off the page;
• differentiability, which means there are no sharp corners in the graph.

Continuity

The intuition of continuity is that you can draw the graph of the function in one piece without lifting your pen off the page.

Continuity at A Point

When we talk about continuity at a point, we want a stream of points \((x, f(x))\) which get closer and closer to the point \((a, f(a))\). In other words, as \(x \to a\) we need \(f(x) \to f(a)\). We can now give a proper definition:

A function \(f\) is continuity at \(x = a\) if \(\lim_{x \to a} f(x) = f(a)\)

So if a function is continuity, it requires:

1. The two-side limit \(\lim_{x \to a}f(x)\) exists(and is finite);
2. The function is defined at \(x = a\); that is, \(f(a)\) exists(and is finite);
3. The two above quantities are equal: \(\lim_{x \to a}f(x) = f(a)\)

Continuity on An Interval

We now know what it means for a function to be continuous at a single point, we extend this definition and say that a function \(f\) is continuous on the interval \((a, b)\) if it is continuous at every point in the interval.

We say a function \(f\) is continuous on \([a, b]\) if:

1. the function \(f\) is continuous at every point in \((a, b)\);
2. the function \(f\) is right-continuous at \(x = a\), that is, \(\lim_{x \to a^+}f(x)\) exists(and is finite), \(f(a)\) exists, and these two quantities are equal;
3. the function \(f\) is left-continuous at \(x = a\), that is, \(\lim_{x \to b^-}f(x)\) exists(and is finite), \(f(b)\) exists, and these two quantities are equal. ;

The Intermediate Value Theorem(IVT)

Knowing that a function is continuous brings some benefits:

1. Intermediate Value Theoram(IVT);
2. Max-Min Theorem.

We can state the Intermediate Value Theorem as:

If \(f\) is continuous on \([a, b]\) and \(f(a) < 0\) and \(f(b) > 0\), then there is at least one number \(c\) in the interval \((a, b)\) such that \(f(c) = 0\). The same is true if instread \(f(a) > 0\) and \(f(b) < 0\).

Maxima and Minima of Continuous Functions

The Max-Min Theorem can be stated as:

If \(f\) is continuous on \([a, b]\), then \(f\) has at least one maximum and one minimum on \([a, b]\).

Differentiability

The differentiablility essentially means that the function has a derivative.

Instantaneous Velocity

How can we measure the velocity of the car at a given instant? The idea is to take the average velocity of the car over smaller and smaller time periods.

Suppose that \(u\) is a short time later than \(t\), let's write \(v_{t \leftrightarrow u}\) to mean the average velocity of the car during the time interval beginning at time \(t\) and ending at time \(u\). Now we just push \(u\) closer and closer to \(t\):

\[ v_{t \leftrightarrow u} = \frac{P_u - P_t} {u - t} = \frac{f(u) - f(t)} {u - t} \]

Notice that the denominator \(u - t\) is the length of time involved, we can just take a limit as \(u \to t\):

\[ v_{t} = \lim_{u \to t} \frac{f(u) - f(t)} {u - t} \]

Since \(u\) is very close to \(t\), we can just write the equotion as:

\[ v_{t} = \lim_{h \to 0} \frac{f(t + h) - f(t)} {h} \]

Tanget Lines

We pick a number \(z\) which is close to \(x\) and plot the point \((z, f(z))\) on the curve and draw the graph:

Since the slope is the rise over the run, the slope of the dashed line is:

\[ \frac{f(z) - f(x)} {z - x} \]

Let's set \(h = z - x\) then we see that as \(z \to x\), we have \(h \to 0\), so we also have:

\[ slope\_of\_tangent\_line\_through (x, f(x)) = \lim_{h \to 0} \frac{f(x + h) - f(x)} {h} \]

The Derivative Function

In the following picture, I've drawn in the tangent lines through three different points on the curve:

These lines have different slopes. That is, the slope of the tangent line depends on which value of \(x\) you start with. Another way of saying this is that the slope of the tangent line through \((x, f(x))\) is itself a function of \(x\). This function is called the derivative of \(f\) and is witten as \(f'\):

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)} {h} \]

If \(y = f(x)\), then you can write \(\frac{dy}{dx}\) instead of \(f'(x)\). For example, if \(y = x^2\), then \(\frac{dy}{dx} = 2x\). In fact, if you replace \(y\) by \(x^2\), you get a variety of different ways of expressing the same thing:

\[ f'(x) = \frac{dy}{dy}{dx} = \frac{d(x^)} {dy} = \frac{d}{dx}(x^2) = 2x \]

Differentiability and Continuity

Now it's time to relate the two big concepts in this chapter:

If a function \(f\) is differentiable at \(x\), then it's continuous at \(x\).

To prove this claim, we have known that \(f\) is continuous at \(x\):

\[ \lim_{u \to x} f(u) = f(x) \]

as \(u \to x\) we can also replace above equation as:

\[ \lim_{h \to 0} f(x + h) = f(x) \]

Now we are aware of our destination, let's start with what we actually know that \(f\) is differentiable at \(x\), which means that \(f'(x)\) exists:

\[ \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} = f'(x) \]

\[ \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \times h = \lim_{h \to 0}f'(x) \times h \]

\[ \lim_{h \to 0} f(x + h) - f(x) = \lim_{h \to 0}f'(x) \times h = 0 \]

so we got:

\[ \lim_{h \to 0} f(x + h) = f(x) \]