SOLVING CALCULUS PROBLEMS

Functions and Limits

Programming Challenges

The following challenges build on what you’ve learned in this notebook. You cand find sample solutions in the solutions notebook.

#1: Verify the Continuity of a Function at a Point

A necessary, but not sufficient, condition for a function to be differentiable at a point is that it must be continuous at that point. That is, the function must be defined at that point and its left-hand limit and right-hand limit must exist and be equal to the value of the function at that point. If $f(x)$ is the function and $x = a$ is the point we are interested in evaluating, this is mathematically stated as

$$\lim_{x\to a^+}f(x) = \lim_{x\to a^-}f(x) = f(a)$$

Your challenge here is to write a program that will (1) accept a single-variable function and a value of that variable as inputs and (2) check whether the input function is continuous at the point where the variable assumes the value input.

Here is a sample working of the completed solution:

Enter a function in one variable: 1/x Enter the variable: x Enter the point to check the continuity at: 1 1/x is continuous at 1.0

The function $1/x$ is discontinuous at 0, so let’s check that:

Enter a function in one variable: 1/x Enter the variable: x Enter the point to check the continuity at: 0 1/x is not continuous at 0.0



In [1]:

    
import sympy



In [2]:

    
def check_continuity(f, var, value):
    '''
    Checks if the function f in the variable var is continuous in value by checking if
    right limit = left limit = f(value)
    '''
    x = sympy.Symbol(var)
    left = sympy.Limit(f, x, value, dir='-').doit()
    right = sympy.Limit(f, x, value, dir='+').doit()
    calc = sympy.sympify(f).subs({x: value})
    if left == right == calc:
        return True
    else:
        return False



In [3]:

    
check_continuity('1/x', 'x', 0)









    Out[3]:





False



In [4]:

    
# Ask user for inputs
f = input('Enter a function in one variable:')
var = input('Enter the variable:')
value = input('Enter the point to check the continuity at:')
# Check if function is continuous
is_continuous = check_continuity(f, var, value)
if is_continuous:
    print('{} is continuous at {}'.format(f, value))
else:
    print('{} is not continuous at {}'.format(f, value))









    



Enter a function in one variable:1/x
Enter the variable:x
Enter the point to check the continuity at:0
1/x is not continuous at 0