Explain why the function is discontinuous at the given number a = 1 ?

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2 Answers
Jun 4, 2018

The limit of the first piece as x approaches 1 from the negative side is:

lim_(x to 1^-) e^x = e ~~ 2.718

We do need to take a limit for the second piece; we can just evaluate it at 1:

1^2 = 1

The function is discontinuous because the two pieces do not have the same y value at 1;

Jun 4, 2018

See below

Explanation:

f(x) is a piecewise function defined above.

To show that f(x) is discontinuous at a given point we can examine the behaviour of the function at that point.

In this case f(1) = 1^2 = 1 because f(x) = x^2: x in [1, +oo)

Now let's consider f(1-delta) for some arbitrarily small delta >0

In this case f(1-delta) = e^(1-delta) because (1-delta) < 1 forall delta >0

So as delta -> 0 then f(1^-) -> e

Since f(1) != f(1^-) -> f(x) is discontinuous at x=1

[NB: f(1^-) means the limit of f(x) as x -> 1 from below.]