No. f is a function that satisfies the functional equation f(x_1 + x_2) = f(x_1) + f(x_2). You're told that f is continuous at 0, and you're supposed to use this to conclude that f is continuous everywhere.Math_Geek said:
Math_Geek said:
Continuity of a function on the reals means that the function has no sudden jumps or breaks and can be drawn without lifting the pen from the paper. In other words, the function has a smooth and continuous graph.
To prove continuity of a function on the reals, we need to show that the limit of the function at a certain point is equal to the value of the function at that point. This can be done using the epsilon-delta definition of continuity or by using the intermediate value theorem.
Continuity and differentiability are related but different concepts. A function is continuous if it has no sudden jumps or breaks, while a function is differentiable if it has a well-defined derivative at every point. A function can be continuous but not differentiable, but if a function is differentiable, it must also be continuous.
The continuity of a function on the reals ensures that its graph is smooth and has no gaps or sharp turns. This means that the function can be drawn without lifting the pen from the paper. If a function is not continuous, its graph may have breaks or jumps, making it difficult to draw.
The concept of continuity of functions on the reals is important in many fields, such as physics, engineering, and economics. It is used to model real-world phenomena and make predictions based on mathematical models. For example, the continuity of a function representing the motion of an object can help predict its future position and velocity.