Calculus

Limits and Continuity of Functions

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Continuity of Functions

Heine Definition of Continuity

A real function \(f\left( x \right)\) is said to be continuous at \(a \in \mathbb{R}\) (\(\mathbb{R}-\) is the set of real numbers), if for any sequence \(\left\{ {{x_n}} \right\}\) such that

\[\lim\limits_{n \to \infty } {x_n} = a,\]

it holds that

\[\lim\limits_{n \to \infty } f\left( {{x_n}} \right) = f\left( a \right).\]

In practice, it is convenient to use the following three conditions of continuity of a function \(f\left( x \right)\) at point \(x = a:\)

Cauchy Definition of Continuity \(\left(\varepsilon \text{-} \delta -\right.\) Definition)

Consider a function \(f\left( x \right)\) that maps a set \(\mathbb{R}\) of real numbers to another set \(B\) of real numbers. The function \(f\left( x \right)\) is said to be continuous at \(a \in \mathbb{R}\) if for any number \(\varepsilon \gt 0\) there exists some number \(\delta \gt 0\) such that for all \(x \in \mathbb{R}\) with

\[\left| {x - a} \right| \lt \delta ,\]

the value of \(f\left( x \right)\) satisfies:

\[\left| {f\left( x \right) - f\left( a \right)} \right| \lt \varepsilon .\]

Definition of Continuity in Terms of Differences of Independent Variable and Function

We can also define continuity using differences of independent variable and function. The function \(f\left( x \right)\) is said to be continuous at the point \(x = a\) if the following is valid:

\[\lim\limits_{\Delta x \to 0} \Delta y = \lim\limits_{\Delta x \to 0} \left[ {f\left( {a + \Delta x} \right) - f\left( a \right)} \right] = 0,\]

where \(\Delta x = x - a.\)

All the definitions of continuity given above are equivalent on the set of real numbers.

A function \(f\left( x \right)\) is continuous on a given interval, if it is continuous at every point of the interval.

Continuity Theorems

Theorem 1.

Let the function \(f\left( x \right)\) be continuous at \(x = a\) and let \(C\) be a constant. Then the function \(Cf\left( x \right)\) is also continuous at \(x = a.\)

Theorem 2.

Let the functions \({f\left( x \right)}\) and \({g\left( x \right)}\) be continuous at \(x = a\). Then the sum of the functions \({f\left( x \right)} + {g\left( x \right)}\) is also continuous at \(x = a.\)

Theorem 3.

Let the functions \({f\left( x \right)}\) and \({g\left( x \right)}\) be continuous at \(x = a.\) Then the product of the functions \({f\left( x \right)}{g\left( x \right)}\) is also continuous at \(x = a.\)

Theorem 4.

Let the functions \({f\left( x \right)}\) and \({g\left( x \right)}\) be continuous at \(x = a\). Then the quotient of the functions \(\frac{{f\left( x \right)}}{{g\left( x \right)}}\) is also continuous at \(x = a\) assuming that \({g\left( a \right)} \ne 0.\)

Theorem 5.

Let \({f\left( x \right)}\) be differentiable at the point \(x = a.\) Then the function \({f\left( x \right)}\) is continuous at that point.

Remark: The converse of the theorem is not true, that is, a function that is continuous at a point is not necessarily differentiable at that point.

Theorem 6 (Extreme Value Theorem).

If \({f\left( x \right)}\) is continuous on the closed, bounded interval \(\left[ {a,b} \right]\), then it is bounded above and below in that interval. That is, there exist numbers \(m\) and \(M\) such that

\[m \le f\left( x \right) \le M\]

for every \(x\) in \(\left[ {a,b} \right]\) (see Figure \(1\)).

Extreme value theorem
Figure 1.

Theorem 7 (Intermediate Value Theorem).

Let \({f\left( x \right)}\) be continuous on the closed, bounded interval \(\left[ {a,b} \right]\). Then if \(c\) is any number between \({f\left( a \right)}\) and \({f\left( b \right)}\), there is a number \({x_0}\) such that

\[f\left( {{x_0}} \right) = c.\]

The intermediate value theorem is illustrated in Figure \(2.\)

Intermediate value theorem
Figure 2.

Continuity of Elementary Functions

All elementary functions are continuous at any point where they are defined.

An elementary function is a function built from a finite number of compositions and combinations using the four operations (addition, subtraction, multiplication, and division) over basic elementary functions. The set of basic elementary functions includes:

  1. Algebraical polynomials
    \[A{x^n} + B{x^{n - 1}} + \ldots + Kx + L;\]
  2. Rational fractions
    \[\frac{{A{x^n} + B{x^{n - 1}} + \ldots + Kx + L}}{{M{x^m} + N{x^{m - 1}} + \ldots + Tx + U}};\]
  3. Power functions \({x^p}\);
  4. Exponential functions \({a^x}\);
  5. Logarithmic functions \({\log_a}x\);
  6. Trigonometric functions
    \[\sin x,\; \cos x,\; \tan x,\; \cot x,\; \sec x,\; \csc x;\]
  7. Inverse trigonometric functions
    \[\arcsin x,\; \arccos x,\; \arctan x,\; \text{arccot}\,x,\; \text{arcsec}\,x,\; \text{arccsc}\,x;\]
  8. Hyperbolic functions
    \[\sinh x,\; \cosh x,\; \tanh x,\; \coth x,\; \text{sech}\,x,\; \text{csch}\,x;\]
  9. Inverse hyperbolic functions
    \[\text{arcsinh}\,x,\; \text{arccosh}\,x,\; \text{arctanh}\,x,\; \text{arccoth}\,x,\; \text{arcsech}\,x,\; \text{arccsch}\,x.\]

Solved Problems

Click or tap a problem to see the solution.

Example 1

Using the Heine definition, prove that the function \[f\left( x \right) = {x^2}\] is continuous at any point \(x = a.\)

Example 2

Using the Heine definition, show that the function \[f\left( x \right) = \sec x\] is continuous for any \(x\) in its domain.

Example 1.

Using the Heine definition, prove that the function \[f\left( x \right) = {x^2}\] is continuous at any point \(x = a.\)

Solution.

Using the Heine definition we can write the condition of continuity as follows:

\[\lim\limits_{\Delta x \to 0} f\left( {a + \Delta x} \right) = f\left( a \right)\;\;\text{or}\;\;\lim\limits_{\Delta x \to 0} \left[ {f\left( {a + \Delta x} \right) - f\left( a \right)} \right] = \lim\limits_{\Delta x \to 0} \Delta y = 0,\]

where \(\Delta x\) and \(\Delta y\) are small numbers shown in Figure \(3.\)

Quadratic function
Figure 3.

At any point \(x = a:\)

\[f\left( a \right) = {a^2},\;\;f\left( {a + \Delta x} \right) = {\left( {a + \Delta x} \right)^2}.\]

So that

\[\Delta y = f\left( {a + \Delta x} \right) - f\left( a \right) = \left( {a + \Delta x} \right)^2 - {a^2} = \cancel{a^2} + 2a\Delta x + {\left( {\Delta x} \right)^2} - \cancel{a^2} = 2a\Delta x + {\left( {\Delta x} \right)^2}.\]

Example 2.

Using the Heine definition, show that the function \[f\left( x \right) = \sec x\] is continuous for any \(x\) in its domain.

Solution.

The secant function \(f\left( x \right) = \sec x = {\frac{1}{{\cos x}}}\) has domain all real numbers \(x\) except those of the form

\[x = \frac{\pi }{2} + k\pi ,\;\;k = 0, \pm 1, \pm 2, \ldots ,\]

where cosine is zero.

Let \(\Delta x\) be a differential of independent variable \(x.\) Find the corresponding differential of function \(\Delta y.\)

\[\Delta y = \sec \left( {x + \Delta x} \right) - \sec x = \frac{1}{{\cos \left( {x + \Delta x} \right)}} - \frac{1}{{\cos x}} = \frac{{\cos - \cos \left( {x + \Delta x} \right)}}{{\cos \left( {x + \Delta x} \right)\cos x}} = \frac{{ - 2\sin \left( {x + \frac{{\Delta x}}{2}} \right)\sin \left( { - \frac{{\Delta x}}{2}} \right)}}{{\cos \left( {x + \Delta x} \right)\cos x}} = \frac{{2\sin \left( {x + \frac{{\Delta x}}{2}} \right)\sin \frac{{\Delta x}}{2}}}{{\cos \left( {x + \Delta x} \right)\cos x}}.\]

Calculate the limit as \(\Delta x \to 0.\)

\[\lim\limits_{\Delta x \to 0} \Delta y = \lim\limits_{\Delta x \to 0} \frac{{2\sin \left( {x + \frac{{\Delta x}}{2}} \right)\sin \frac{{\Delta x}}{2}}}{{\cos \left( {x + \Delta x} \right)\cos x}} = \lim\limits_{\Delta x \to 0} \frac{{2\sin \left( {x + \frac{{\Delta x}}{2}} \right)}}{{\cos \left( {x + \Delta x} \right)\cos x}} \cdot \lim\limits_{\Delta x \to 0} \sin \frac{{\Delta x}}{2} = \frac{{2\sin x}}{{{{\cos }^2}x}} \cdot 0 = 0.\]

This result is valid for for all \(x\) except the roots of the cosine function:

\[x = \frac{\pi }{2} + k\pi ,\;\;k = 0, \pm 1, \pm 2, \ldots\]

Hence, the range of continuity and the domain of the function \(f\left( x \right) = \sec x\) fully coincide.

See more problems on Page 2.

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