a is simply the point under consideration. More verbosely the question could ask "Let a=0. Decide whether the following functions are continuous at x=a ...". So, you simply need to examine the continuity of the f(x) at the origin.Fairy111 said:Homework Statement
Decide whether the following functions are continuous at a=0
Homework Equations
f(x)=x^2 if x<0 and f(x)=sinx if x>=0
The Attempt at a Solution
I don't really understand where the 'a' comes in the functions? Some help/hints as to how to start off this question would be much appreciated.
Fairy111 said:ok, thankyou.
So I am looking at the left hand part of the parabola of x^2, which is when x<0.
It stops at x=0, I am not really sure what to say about the continuity.
A function is said to be continuous at a specific point if the limit of the function at that point exists and is equal to the value of the function at that point.
To determine if a function is continuous at a=0, we need to evaluate the limit of the function at a=0 and check if it is equal to the value of the function at a=0. If the limit and the value are equal, then the function is continuous at a=0.
The three conditions for a function to be continuous at a=0 are: (1) the function is defined at a=0, (2) the limit of the function at a=0 exists, and (3) the limit is equal to the value of the function at a=0.
No, a function cannot be continuous at a=0 if it is not defined at a=0. The function must be defined at the point of interest for it to be considered continuous at that point.
A function being continuous at a=0 indicates that there are no breaks or jumps in the graph of the function at that point. This means that the function is smooth and well-behaved at that point, and we can use it to make predictions and solve problems with confidence.